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<font size=+2>

Numerical Integration
</font>

 &nbsp;

<a href="#docB037">
Update 2010-02-09
</a>

<br>
<a href="#NumIntegral">Integration</a>
　
<a href="#differentiate">Differentiation</a>
　
<a href=integral.htm#docB007>
How to use
</a>
　
<a href=integral.htm#docB030>
Link
</a>
　
<br>

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XYGraph v2.3 - web page graph
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donate<!--9801031950 use donate ##9829;##9835; ♥♫-->
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get code</a>

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2006-06-12-18-21 download
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C:\$fm\js\xygraph\XYGraph.zip
2006-06-19-08-21 record
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<br>

<a href=http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/CSMC_index.html>
The Cauchy-Schwarz Master Class
</a>
 &nbsp;
<a href=http://www-stat.wharton.upenn.edu/~steele/index.html>
J. Michael</a>
 
<a href=http://www.wharton.upenn.edu/faculty/steele.html>
Steele</a>
 &nbsp;
<a href=http://www.amazon.com/review/product/052154677X/ref=cm_cr_dp_all_helpful?%5Fencoding=UTF8&coliid=&showViewpoints=1&colid=&sortBy=bySubmissionDateDescending>
★★★★★
</a><!--9806081839 add link-->


<br>
<font color=red>
This file is personal home work. No one
<br>
proofread. Cannot promise correctness.
<br>
If you suspect any view point wrong, 
<br>
please ask a math expert near by.
<br>
Freeman 2009-06-19-10-46</font>
<br>
Please use MSIE browser to read this file.
<br>
Did not test other browser. This file is
<br> <!--9806081846-->
written under MSIE 6.0

<br>
<span id="tuteLink1"></span>
<br>
<script src="http://freeman2.com/rocsitee.js" language="javascript"></script>
<br>

<pre><font size=+2>
<a name="docA001">&lt;a name="docA001"&gt;</a>
2010-01-30-12-11 start
2009-06-09-10-09 begin read
The Cauchy-Schwarz Master Class
written by Prof. J. Michael Steele
ISBN 978-0-521-54677-5
2010-01-22 <a href=http://freeman2.com/tute0027.htm>done chapter six</a>
<a href=tute0027.htm>Convexity</a> ~~ The Third Pillar
<a name="docA002">&lt;a name="docA002"&gt;</a>
Start chapter seven 
Integral Intermezzo
LiuHH need a program to calculate
function integration value to aid
reading chapter seven. Before
begin chapter seven, write a
program integral.htm as a tool.
Program is done. Upload integral.htm
let every one use this program.
2010-01-30-12-18 stop

<hr>

</font></pre>


<a name="NumIntegral">&lt;a name="NumIntegral"&gt;</a>
 2010-01-28-20-20

<br>
■　Numerical Integration　(<a href="#differentiate">Differentiation</a>)

<br>
<font color=red><b>
Output may contain error, Please verify first
<br>
Program environment is MSIE 6.0, please use MSIE
<br>
<!--9901132047-->If you save this file integral.htm to your computer
<br>
and open local integral.htm, it can not draw figure.
<br>
You need also save http://freeman2.com/jsgraph2.js
<br>
to your computer stay in same folder as integral.htm.
</b></font>

<br>

x min:
  <INPUT id=QBboxXmin10
   value="-6" size=3
  >
, x max:
  <INPUT id=QBboxXmax10
   value="6" size=3
  >
; y min:
  <INPUT id=QBboxYmin10
   value="-30" size=3
  >
, y max:
  <INPUT id=QBboxYmax10
   value="30" size=3
  >
;

<br>
x min, x max, y min, y max is coordinate axis range 

<br>

Drawing board size W: 
  <INPUT id=QBboxWidth10
   value="300" size=5
   onchange="javascript:if(QBboxWidth10.value<100)QBboxWidth10.value=100; else if(QBboxWidth10.value>800)QBboxWidth10.value=800;"
  >
H: 
  <INPUT id=QBboxHeight10
   value="300" size=5
   onchange="javascript:if(QBboxHeight10.value<100)QBboxHeight10.value=100; else if(QBboxHeight10.value>800)QBboxHeight10.value=800;"
  >

<br>
<a name="IntegralBgn"></a>
<font color=red><b>
Next 7 lines are for integration. Variable must be x.
</b></font>
<br>
Example: 
<input onclick="javascript:QBboxf0Int.value=
'x*x*x-2*x*x+7*x-5';
QBboxf0xBgn.value='1';
QBboxf0xEnd.value='3';
QBboxf0xStp.value='4';
RadioNCF[1].checked=1; //[1] goto '2' in '1,3,4,2,0'
QBspanIntegral.innerHTML=
integralf1('x*x*x-2*x*x+7*x-5',1,3,4,2,0);"
 title="SBN 471-13507-0 page 74 example"
 type="button" value="1" /><!--9901291528-->

  <input onclick="javascript:QBboxf0Int.value=
'sin(x)';
QBboxf0xBgn.value='0';
QBboxf0xEnd.value='PI';
QBboxf0xStp.value='10';
RadioNCF[5].checked=1; //[5] goto '6' in '0,3.141592653589793,10,6,0'
QBspanIntegral.innerHTML=
integralf1('sin(x)',0,3.141592653589793,10,6,0);"
 title="x in [0,PI]. 'PI' can be used."
 type="button" value="2" />
<!--9901291530
calling
integralf1('sin(x)',0,3.141592653589793,10,6,0)
parameters are
'sin(x)'  function definition
0         x begin value
3.14159   x end value
10        step=10, from xbgn to xend,
          divide to 10 intervals, each
          interval do one Newton-Cotes
          formula evaluation.
6         Newton-Cotes formula n=6 method
0         short output to page <span ...>
(1  is longer output to box 21)
2010-01-29-19-02
-->

  <input onclick="javascript:QBboxf0Int.value=
'1/(1+x*x)';
QBboxf0xBgn.value='-4';
QBboxf0xEnd.value='+4';
QBboxf0xStp.value='1';
RadioNCF[5].checked=1; //[5] goto '6' in '-4,4,1,6,0'
QBspanIntegral.innerHTML=
integralf1('1/(1+x*x)',-4,4,1,6,0);"
 title="ISBN 0-471-02985-8 page 229 example. 2*atan(4)=2.651635327336065"
 type="button" value="3" /><!--9901291615-->

  <input onclick="javascript:QBboxf0Int.value=
'sin(x)*sqrt(1-x*x)';
QBboxf0xBgn.value='0';
QBboxf0xEnd.value='1';
QBboxf0xStp.value='1';
RadioNCF[5].checked=1; //[5] goto '6' in '0,1,1,6,0'
QBspanIntegral.innerHTML=
integralf1('sin(x)*sqrt(1-x*x)',0,1,1,6,0);"
 title="ISBN 0-471-02985-8 page 268 line -2. dg(x)/dx singular at x=1"
 type="button" value="4" /><!--9901291646-->

  <input onclick="javascript:QBboxf0Int.value=
'2*sin(1-x*x)*x*x*sqrt(2-x*x)';
QBboxf0xBgn.value='0';
QBboxf0xEnd.value='1';
QBboxf0xStp.value='1';
RadioNCF[5].checked=1; //[5] goto '6' in '0,1,1,6,0'
QBspanIntegral.innerHTML=
integralf1('2*sin(1-x*x)*x*x*sqrt(2-x*x)',0,1,1,6,0);"
 title="ISBN 0-471-02985-8 page 268 line -2. dg(x)/dx OK at x=1"
 type="button" value="5" /><!--9901291650-->

  <input onclick="javascript:QBboxf0Int.value=
'1/x';
QBboxf0xBgn.value='x=0;exp(x)';
QBboxf0xEnd.value='E';
QBboxf0xStp.value='2';
RadioNCF[5].checked=1; //[5] goto '6' in '1,6,0'
QBspanIntegral.innerHTML=
integralf1('1/x',1,2.718281828459045,2,6,0);"
 title="From x=1 to x=E Integral of 1/x = 1"
 type="button" value="6" /><!--9902022328-->


  <input onclick="javascript:QBboxf0Int.value=
'exp(-x)';
QBboxf0xBgn.value='exp(0)-sin(1)*sin(1)-cos(1)*cos(1)';
QBboxf0xEnd.value='1.E3';
QBboxf0xStp.value='200';
RadioNCF[5].checked=1; //[5] goto '6' in '1,6,0'
QBspanIntegral.innerHTML=
integralf1('exp(-x)',0,1000,200,6,0);"
 title="Exact solution=1, need large steps."
 type="button" value="7" /><!--9902052354-->


 &nbsp; 
<a href="#box21">debug</a> 
 &nbsp; 
<a href="#docB007">help1</a>
 &nbsp; 
  <input onclick="
QBboxfunc0x.value=QBboxf0Int.value;
QBboxfunc1x.value=QBboxfunc2x.value=
QBboxfunc3x.value=''"
 type="button" value="g2f" /><!--9901301400-->

 &nbsp; 
<a href="#docB028">help2</a>

<br>

g(x)<!-- value="x*x*x-2*x*x+7*x-5"
 SBN 471-13507-0 page 74 example-->
  <INPUT id=QBboxf0Int
   value="x*x*x-2*x*x+7*x-5"
   size=60
  >

<input onclick="QBboxf0Int.value=''"
 type="button" value="d" />

<br>

<!--9901291455 distinguish 
integral g(x) A //output to QBspanIntegral.innerHTML
from 
integral g(x) B //output to box21

click button [integral g(x) A] 
use
QBspanIntegral.innerHTML=integralf1(parameters...)

click button [integral g(x) B] 
use
QBboxd21.value=integralf1(parameters...)
-->

<font color=red><b>
RUN</b></font>
  <input onclick="javascript:var ncfPick=-1;for(var w0=0;w0<RadioNCF.length;w0++)if(RadioNCF[w0].checked){ncfPick=w0;break;}ncfPick++;QBspanIntegral.innerHTML=integralf1(QBboxf0Int.value,QBboxf0xBgn.value,QBboxf0xEnd.value,QBboxf0xStp.value,ncfPick,0)"
 type="button" value="integral g(x) A"
 />
 &nbsp; 
<a href="#docB015">
steps</a>
  <INPUT id=QBboxf0xStp
   value="5";
   size=3
  >

<input onclick="QBboxf0xStp.value=''"
 type="button" value="ds" />

<input onclick="QBboxf0xBgn.value=QBboxf0xEnd.value=''"
 type="button" value="dx" />

<input onclick="QBspanIntegral.innerHTML='<br>'"
 type="button" value="da" />

<br>

<!--9902091619 use larger box for xbgn, xend-->
x-bgn
 <INPUT id=QBboxf0xBgn
   value="1";
   size=22
  >
x-end
  <INPUT id=QBboxf0xEnd
   value="3";
   size=22
  >

<br>

Newton-Cotes formula n=1 <!--9901281831 use radio-->
 <INPUT name="RadioNCF" type="radio">, 
2<INPUT name="RadioNCF" type="radio" checked >, 
3<INPUT name="RadioNCF" type="radio">, 
4<INPUT name="RadioNCF" type="radio">, 
5<INPUT name="RadioNCF" type="radio">, 
6<INPUT name="RadioNCF" type="radio">

<br>

<font color=red><b>
<span id=QBspanIntegral>
draw f(x) must use x as independent variable,
<br>
draw x(t),y(t) must use t as independent variable.
</span>
</b></font>
<br>

<a name="clickFig0605"></a>
<a name="GraphBgn"></a>
Graph title: 
  <INPUT id=QBboxTitle10
   value="input title here; below graph, above integral.";
   size=50
  >

<input onclick="QBboxTitle10.value=''"
 type="button" value="d" />

<br>

x/t begin 
  <INPUT id=QBboxXbegin
   value="-6" size=20
  >
, x/t end
  <INPUT id=QBboxXend
   value="6" size=20
  >

<input onclick="QBboxXbegin.value=QBboxXend.value=''"
 type="button" value="d" />

<br>

Draw curve in 
  <INPUT id=QBboxSteps
   value="180" size=3
  >
 steps

<input onclick="QBboxSteps.value=''"
 type="button" value="d" />

　<a href="#JSMathList">math function</a>

　<a href="#docB035">x^5</a>

<br><font color=red>
■</font>f0(x)<!--9901241407 define f(x)-->
  <INPUT id=QBboxfunc0x
   value="gatef(x,-7,0)*(9*pow(abs(x),4/3)/4+9*pow(abs(x),1/3))+gatef(x,0,7)*(9*pow(abs(x),4/3)/4-9*pow(abs(x),1/3))";
   size=50
  >
 <font color=red>x1(t)</font> 

<input onclick="QBboxfunc0x.value=''"
 type="button" value="d" />


<br><font color=black>
■</font>f1(x)
  <INPUT id=QBboxfunc1x
   value="gatef(x,-7,0)*(-3*pow(abs(x),1/3)-3*pow(abs(x),-2/3))+gatef(x,0,7)*(3*pow(abs(x),1/3)-3*pow(abs(x),-2/3))";
   size=50
  >

 <font color=red>y1(t)</font> 

<input onclick="QBboxfunc1x.value=''"
 type="button" value="d" />

<br><font color=blue>
■</font>f2(x)
  <INPUT id=QBboxfunc2x
   value="gatef(x,-7,0)*(pow(abs(x),-2/3)-2*pow(abs(x),-5/3))+gatef(x,0,7)*(pow(abs(x),-2/3)+2*pow(abs(x),-5/3))";
   size=50
  >

 <font color=blue>x2(t)</font> 

<input onclick="QBboxfunc2x.value=''"
 type="button" value="d" />


<br><font color=purple>
■</font>f3(x)
  <INPUT id=QBboxfunc3x
   value="";
   size=50
  >

 <font color=blue>y2(t)</font> 

<input onclick="QBboxfunc3x.value=''"
 type="button" value="d" />

<input onclick="QBboxfunc0x.value=QBboxfunc1x.value=QBboxfunc2x.value=QBboxfunc3x.value=''"
 type="button" value="dA" />



<br>

  <input onclick="javascript:generalDrawf(1)"
 type="button" value="Draw f0(x)
 to f3(x)"
 title="f0(x) to f3(x) must use 'x' as variable"
 />

 &nbsp; 

  <input onclick="javascript:generalDrawf(101)"
 type="button" value="x1(t), y1(t)
 x2(t), y2(t)"
 title="x1(t) y1(t) must use 't' as variable"
 />

 &nbsp; 

  <input onclick="QBdivGraph10.innerHTML=QBspanEqnDoc10.innerHTML=''"
 type="button" value="wipe" />

<br>
Above big button run user equation. Below run program equation.

<br>
f(x) 
  <input onclick="javascript:generalDrawf(11)"
 type="button" value="11" />

  <input onclick="javascript:generalDrawf(12)"
 type="button" value="12" />

 p1 <INPUT id=QBboxPara01
   value="1";
   size=5
  ><!--9901261529-->

 p2 <INPUT id=QBboxPara02
   value="2";
   size=5
  >

 ; x(t),y(t) 

  <input onclick="javascript:generalDrawf(54)"
 type="button" value="54" />

  <input onclick="javascript:generalDrawf(55)"
 type="button" value="55" />


<br>
<font color=red>
<span id=QBspanAlert10>

</span>
</font>
<br>

<div id=QBdivGraph10></div>

<br>
<span id=QBspanEqnDoc10> <!--98111108361-->
</span>

<br>

<a name="box21">&lt;a name="box21"&gt;</a>
Click "integral g(x) B", debug output to box21.

<br>

Box 21,<!--9901072050 use box21-->

<INPUT onclick='javascript:window.clipboardData.setData("Text",document.getElementById("QBboxd21").value.toString())'
 type=button value="copy21">
<INPUT onclick='document.getElementById("QBboxd21").value=""'
 type=button value="del21">

<!--9901291455 distinguish 
click button [integral g(x) A] 
use
QBspanIntegral.innerHTML=integralf1(parameters...,0)

click button [integral g(x) B] 
use
QBboxd21.value=integralf1(parameters...,1)
-->
  <input onclick="javascript:var ncfPick=-1;for(var w0=0;w0<RadioNCF.length;w0++)if(RadioNCF[w0].checked){ncfPick=w0;break;}ncfPick++;QBboxd21.value=integralf1(QBboxf0Int.value,QBboxf0xBgn.value,QBboxf0xEnd.value,QBboxf0xStp.value,ncfPick,1)"
 type="button" value="integral g(x) B"
 />

 &nbsp; 

<a href="#IntegralBgn">Control Center</a>

<br>
<TEXTAREA
 id=QBboxd21
 name=QBboxd21 rows=6
 cols=60
 >
Box 21 record debug data. 
Please click "integral g(x) B" button.
</TEXTAREA>

<br>
You can not draw other curve here. But you can
<br>
goto <a href="tute0013.htm#ch02c023">[Modify 606]</a> define your equation
<br>
and click [Draw] within yellow stripe. 
<br>
(Do not click [Draw 606]) 2009-10-08-17-08 
<br>


<script lang="javascript">
<!--


var ncfCoef= //9901281755
[ //ncfCoef is used in calculation
[1/2, 1, 1, -1/12],
[1/3, 1, 4, 1, -1/90],
[3/8, 1, 3, 3, 1, -3/80],
[2/45, 7,32,12,32,7,-8/945],
[5/288,19,75,50,50,75,19,-275/12096],
[1/140,41,216,27,272,27,216,41,-9/1400]
];


var ncfCoef2= //9901292155
[ //ncfCoef2 is used in document
'1/2; 1,  1 ; -1/12', 
'1/3; 1,  4,  1 ; -1/90', 
'3/8; 1,  3,  3,  1 ; -3/80', 
'2/45; 7, 32, 12, 32, 7 ; -8/945', 
'5/288; 19, 75, 50, 50, 75, 19 ; -275/12096', 
'1/140; 41, 216, 27, 272, 27, 216, 41 ; -9/1400'
];



/**
function integralf1() use
Newton-Cotes formula n=1 to 6
to find function integration value.
int{f(x)dx}, must use 'x' as 
independent variable.

function integralf1() receive 
six parameters
inArg1=integrand equation definition
inArg2=xbegin value
inArg3=xend value
inArg4=steps value, positive integer
inArg5=ncfchoice value, 1 to 6
inArg6=output method

if(inArg6==0) //9901291451
output to QBspanIntegral.innerHTML
only two lines, use <br> change line

if(inArg6==1) //9901291452
output to box21 for detail
output many lines. Use '\n' change line.

Two example calls are
QBspanIntegral.innerHTML=
integralf1('1/(1+x*x)',-4,4,1,6,0);

QBboxd21.value=
integralf1(QBboxf0Int.value,QBboxf0xBgn.value,QBboxf0xEnd.value,QBboxf0xStp.value,ncfPick,1)

QBboxf0Int, QBboxf0xBg, ... are
web page specific. You can use
different element ID. Must match
right.
9901300548

/**/

function integralf1(inArg1,inArg2,inArg3,inArg4,inArg5,inArg6)
{//9901281800 
//integration function 1st method
var w0,w1,w2,w3,w4;
var msg0='';
var intSum=0; //answer, integration value

var eqnfx=inArg1; //integrand equation definition

//x_begin, x_end domain two ends
var xbgn=parseFloat(eval('with(Math){'+inArg2+'}'));
var xend=parseFloat(eval('with(Math){'+inArg3+'}'));
//"with(Math){...}" main goal is allow
//user input "PI" as domain end value.
//also allow "x=2;sin(x)" or
//"vv=0.5;exp(vv)" as domain end value.

var stps=parseInt(inArg4); 
//if stps=2, whole domain cut to two
//equal interval. Each interval do
//one Newton-Cotes formula integration.

if(isNaN(xbgn))
{return 'xbgn wrong:'+inArg2+'<br>';}
if(isNaN(xend))
{return 'xend wrong:'+inArg3+'<br>';}
if(isNaN(stps))
{return 'step wrong:'+inArg4+'<br>';}
if(stps<=0)
{return 'step wrong:'+inArg4+', must be positive.<br>';}
if(stps>100&&inArg6==1) //9902060019
{inArg6=0; alert('debug output allow at most steps=100\nNow steps='+stps+'\nChange to regular output, no debug.')}

var ncfchoice=inArg5;
if(isNaN(ncfchoice))
{return 'Newton-Cotes formula selection wrong:'+inArg5+'<br>';}
//This page never wrong. But for general
//consideration, include guarding code.
if(ncfchoice<1)ncfchoice=1
else
if(ncfchoice>6)ncfchoice=6

//total stps interval, each interval
//create ncfchoice function values
var xCount=ncfchoice*stps
var xdelta=(xend-xbgn)/xCount;
//xdelta is 'h' in (h/3)*[f(x0)+4*f(x1)+f(x2)]
//function evaluation times=xCount+1

//record function value
var farray=[0,0,0,0,0,0,0];
var oneSum=0;
//in (h/3)*[f(x0)+4*f(x1)+f(x2)]
//oneSum is "[f(x0)+4*f(x1)+f(x2)]"
//(h/3)*oneSum is one section integration 
//value. There are total 'stps' sections.

var fvalue;

//eqnfx use 'x' as independent variable.
//before each "eval('with(Math){'+eqnfx+'}')"
//must update x value
x=xbgn
fvalue=farray[0]=
eval('with(Math){'+eqnfx+'}');

//next line is last error code before correct answer.
//oneSum+=eval(eqnfx); //9901291257 delete
//9901291257 deleted then get correct answer
//integral[x=1,3]{xdx} = 4 right!! 9901291258

var echoDebug=0; //return answer only, no debug detail
if(arguments.length>=6&&inArg6==1)
echoDebug=1; //9901291801, return msg0=debug detail

if(echoDebug) // print debug detail
  {
msg0+=''
+'equation="'+eqnfx+'"\n'
+'ncfchoice=Newton-Cotes formula user choice [1 to 6]\n'
+'program count start from 0, ncfchoice-1='+(ncfchoice-1)+'\n'
+'Program use the following coefficients:\n'
+ncfCoef2[ncfchoice-1]+'\n' //9901292159 use this line
+'First coefficient is for x_interval length.\n'
+'Last coefficient is for error estimation.\n'
+'First is used at integral value evaluation point.\n'
+'Last is not used in program.\n'
+'Middle terms are coefficients for function values.\n'
+'First coefficient modify domain x value.\n'
+'Middle coefficient modify range function values.\n'
+'Simple example:\n'
+'for n=2 (Simpson\'s Rule), h=(x2-x0)/2\n'
+'Integral[x=x0,x2]f(x)dx=\n'
+'(h/3)*[f(x0)+4*f(x1)+f(x2)] - h^5*f(4)(ξ)/90\n'
+'"f(4)" is f(x) differential four times. not f(x=4)\n'
+'Above is polynomial formula for any integrated\n'
+'function. Program store coefficients as next\n'
+'[1/3, 1, 4, 1, -1/90]\n'
+'Program allow user integrate f(x) in many steps.\n'
+'If step=1, whole domain apply Simpson Rule once.\n'
+'If step=2, whole domain divide to two smaller\n'
+'sections, Each section apply Simpson Rule once.\n'
+'Variable oneSum sum [f(x0)+4*f(x1)+f(x2)] which \n'
+'is NOT integral of one section. Missing "(h/3)"\n'
+'(h/3)*oneSum is partial intSum in one step.\n'
+'Add all (h/3)*oneSum get final answer intSum\n'
+'Echo debug information start.\n'
+'\n'
;
  }

//if stps=2, whole domain cut to two
//equal interval. Each interval do
//one Newton-Cotes formula integration.
//
for(w0=0;w0<stps;w0++) // 9901282118 correct
  { //w0=number of steps
//stps=total number of steps
//Assume domain=[1,5], assume 
//Newton-Cotes formula n=2 selected
//assume number of steps=2
//First step calculate x=1,2,3 three
//function values. (Simpson's Rule)
//Second step calculate x=3,4,5 three
//function values. x=3 is previous step
//last evaluation. Use it for this step
//9901300505 write note.
if(w0>0) //first step no previous.
    {
if(echoDebug)
      {
msg0+='\n'
+'Start a new step. New step first function value\n'
+'is previous step last function value as following\n'
+bye09(fvalue)+' &&& ';
      }
farray[0]=fvalue;
//farray[0], [1], [2] store current x=3,4,5
//three function values. fvalue is previous
//step last function value. USE IT. 9901300512
    }

//in (h/3)*[f(x0)+4*f(x1)+f(x2)]
//oneSum is "[f(x0)+4*f(x1)+f(x2)]"
//for n=2, coefficients are
//[1/3, 1, 4, 1, -1/90] = ncfCoef[1]
//ncfCoef[2-1][0] is 1/3 for "(h/3)*"
//ncfCoef[2-1][1] is first 1 for "f(x0)"
//Next line is  fvalue*ncfCoef[2-1][1] //9901300517
oneSum+=fvalue*ncfCoef[ncfchoice-1][1];//9901291216
if(echoDebug)
    {
msg0+='x='+x+'\n';
msg0+='ncfCoef[ncfchoice-1][1]='+ncfCoef[ncfchoice-1][1]+'\n';
msg0+='fvalue='+bye09(fvalue)+' ;;; ';
msg0+='oneSum#='+bye09(oneSum)+'\n\n';
    }

//Above oneSum get value from "f(x0)"
//in "[f(x0)+4*f(x1)+f(x2)]"
//below find 4*f(x1)+f(x2) //9901300520
for(w1=0;w1<ncfchoice;w1++)
    {
//before each "eval('with(Math){'+eqnfx+'}')"
//must update x value
x=xbgn+xdelta*(w0*ncfchoice+w1+1)
//find x value, start from xbgn
//for ncfchoice steps done, there are
//w0*ncfchoice previous xdelta used
//go over used xdelta, "w1+1" is
//current new xdelta. "+1" in "w1+1"
//is current first function value
//which copied from previous last one.
//current first is done. go over it.

farray[w1+1]=
fvalue=eval('with(Math){'+eqnfx+'}');

//9901291246 change from [w1+1] to [w1+2]
oneSum+=fvalue*ncfCoef[ncfchoice-1][w1+2]
//ncfCoef[ncfchoice-1][w1+0] used at 9901291220
//ncfCoef[ncfchoice-1][w1+1] used at 9901291216
//[w1+1] is an error code. [w1+2] OK.
//in [1/3, 1, 4, 1, -1/90]
//[w1+0] is '1/3' it is for (h/3) in
//(h/3)*[f(x0)+4*f(x1)+f(x2)]
//[w1+1] is first '1', it is used at
//"copied from previous last one", then
//[w1+2] is correct '4' in [1/3, 1, 4, 1, -1/90]
//in (h/3)*[f(x0)+4*f(x1)+f(x2)], 'h' is xdelta 
//9901300530

if(echoDebug)
msg0+=''
+'x='+bye09(x)+', '
+'w1='+w1+', '
+'ncfchoice='+ncfchoice+'\n'
+'ncfCoef[ncfchoice-1][w1+2]='+ncfCoef[ncfchoice-1][w1+2]+'\n'
+'fvalue='+bye09(fvalue)+', '
+'oneSum%=old_oneSum+'+ncfCoef[ncfchoice-1][w1+2]
+'*fvalue='+bye09(oneSum)+'\n\n'
    } //for(w1=0;w1<ncfchoice;w1++)

//in (h/3)*[f(x0)+4*f(x1)+f(x2)],
//'h' is xdelta 
//'1/3' is ncfCoef[ncfchoice-1][0]
//'[f(x0)+4*f(x1)+f(x2)]' is oneSum
intSum+=xdelta*ncfCoef[ncfchoice-1][0]*oneSum; //9901291220

if(echoDebug)
    {
msg0+=''
+(ncfchoice+1)+' function values used are next\n';
for(w1=0;w1<=ncfchoice;w1++)
      { //9901291048
msg0+=farray[w1]+'; ';
      }
msg0+='\n'
msg0+=''
+'xdelta='+xdelta+'\n'
+'ncfCoef[ncfchoice-1][0]='+ncfCoef[ncfchoice-1][0]+'\n'
+'oneSum&='+oneSum+'\n'
+'intSum+=xdelta*ncfCoef[ncfchoice-1][0]*oneSum;\n'
+'intSum+=this intSum='+(xdelta*ncfCoef[ncfchoice-1][0]*oneSum)+'\n'
+'upto now intSum='+intSum+'\n' //9901291233
msg0+='\n\n=====\n'; //9901291050
    }

//done one step, reset oneSum
oneSum=0; //9901291221
  } //for(w0=0;w0<stps;w0++)

//intSum is final answer, present to user.
//if answer is 2.0000000000003
//bye09() change answer to 2
//if answer is 6.2999999999999
//bye09() change answer to 6.3
//if answer is 7.3333333333333
//bye09() NOT change answer
//9901300538
intSum=bye09(intSum); //9901292233

if(echoDebug)
  {
msg0+='\n\n';
msg0+='eqnfx='+eqnfx+'\n';
msg0+='xbgn='+xbgn+'\n';
msg0+='xend='+xend+'\n';
msg0+='stps='+stps+'\n';
msg0+='Newton-Cotes formula choice=ncfchoice='+ncfchoice+'\n';
msg0+='Total function evaluation=ncfchoice*stps+1='+(xCount+1)+'\n';//9901282009
msg0+='xdelta=(xend-xbgn)/(ncfchoice*stps)='+xdelta+'\n';
  }

if(echoDebug) //return to box21, long string
  {
return 'Answer : intSum='+intSum+'\n' //9901291222
+msg0; //9901281805
  }
else //return to web page <span> two lines.
return 'From x='+xbgn+' to x='+xend+'<br>'
+'integration value is '+intSum;

} //function integralf1(inArg1) 9901291258_first_OK
//9901300540 done document notes.




function generalDrawf(rtArg1) //9901061920
{

QBspanAlert10.innerHTML='';
var b0,b1,b2,b3,b4,b5,b6,b7,b8,b9;
var ii,jj,kk; 

var fpar01=parseFloat(QBboxPara01.value); //9901251536
var fpar02=parseFloat(QBboxPara02.value);
if(isNaN(fpar01))fpar01=QBboxPara01.value=0;
if(isNaN(fpar02))fpar02=QBboxPara02.value=0;




if(rtArg1==11)
  { //9901251600
QBboxXbegin.value='sin(0)'
QBboxXend.value='a=0;b=2;c=1;cos(a)*pow(b,c)'
QBboxXmin10.value=-2
QBboxXmax10.value=+2
QBboxYmin10.value=-2
QBboxYmax10.value=+2

QBboxfunc0x.value='x*('+fpar02+')/((1-x)*('+fpar01+')+x*('+fpar02+'))'; //9901251533

//QBspanAlert10.innerHTML= //9901251604
QBboxfunc1x.value='('+fpar02+')/((1-x)*('+fpar01+')+x*('+fpar02+'))-x*('+fpar02+')*(-('+fpar01+')+('+fpar02+'))/((1-x)*('+fpar01+')+x*('+fpar02+'))/((1-x)*('+fpar01+')+x*('+fpar02+'))'
QBboxfunc2x.value=
QBboxfunc3x.value='';
QBboxTitle10.value='www.cms.math.ca/cjm/v28/cjm1976v28.0793-0804.pdf';


//return;  //9901251605

  }
else
if(rtArg1==12)
  { 
//9901251148
QBboxXmin10.value=-6
QBboxXmax10.value=6
QBboxYmin10.value=-30
QBboxYmax10.value=30
QBboxXbegin.value=-6
QBboxXend.value=6
QBboxSteps.value=180
QBboxfunc0x.value
='gatef(x,-7,0)*(9*pow(abs(x),4/3)/4+9*pow(abs(x),1/3))+gatef(x,0,7)*(9*pow(abs(x),4/3)/4-9*pow(abs(x),1/3))'
QBboxfunc1x.value
='gatef(x,-7,0)*(-3*pow(abs(x),1/3)-3*pow(abs(x),-2/3))+gatef(x,0,7)*(3*pow(abs(x),1/3)-3*pow(abs(x),-2/3))'
QBboxfunc2x.value
='gatef(x,-7,0)*(pow(abs(x),-2/3)-2*pow(abs(x),-5/3))+gatef(x,0,7)*(pow(abs(x),-2/3)+2*pow(abs(x),-5/3))'
//9901251154

QBboxTitle10.value='marauder.millersville.edu/~bikenaga/calculus/conc/conc.pdf';

  }

//rtArg1>50 is for x(t), y(t)
//must use 't' as independent variable
//must NOT use 'x'. 9901071100
else 
if(rtArg1==54)
  { //9901202050
QBboxXbegin.value='-PI'
QBboxXend.value='+PI'
QBboxSteps.value=180
QBboxXmin10.value=-30
QBboxXmax10.value=+30
QBboxYmin10.value=-30
QBboxYmax10.value=+30
QBboxfunc0x.value='18*cos(t)*cos(2*t)*cos(t)';
QBboxfunc1x.value='18*cos(t)*cos(PI*t)*sin(t)';
QBboxfunc2x.value='13*cos(t)+sin(8*t)';
QBboxfunc3x.value='13*sin(t)+cos(8*t)';
QBboxTitle10.value="";
  }


else 
if(rtArg1==55)
  { //9901202057
QBboxXbegin.value=0.5
QBboxXend.value=+15
QBboxSteps.value=180
QBboxXmin10.value=-2
QBboxXmax10.value=+2
QBboxYmin10.value=-2
QBboxYmax10.value=+2
QBboxfunc0x.value='cos(t)*cos(t)/t';
QBboxfunc1x.value='sin(t)*sin(t)/t';
QBboxfunc2x.value='1*cos(t)*cos(t)*cos(t)';
QBboxfunc3x.value='1*sin(t)*sin(t)*sin(t)';
QBboxTitle10.value="";
  }



//9902050040 add "with(Math)..."
var xbgn=parseFloat(eval('with(Math){'+QBboxXbegin.value+'}'))
var xend=parseFloat(eval('with(Math){'+QBboxXend.value+'}'))
var step5=parseFloat(QBboxSteps.value);

if(isNaN(xbgn))
    { //9901062323
QBspanAlert10.innerHTML=''
+'x begin must be defined. Now it is ['
+QBboxXbegin.value+']';
return;
    }

if(isNaN(xend))
    { //9901062324
QBspanAlert10.innerHTML=''
+'x end must be defined. Now it is ['
+QBboxXend.value+']';
return;
    }

if(xbgn>xend)
    { //9901062326
QBspanAlert10.innerHTML=''
+'x begin must be < x end. Now '
+'x begin='+QBboxXbegin.value
+', x end='+QBboxXend.value;
return;
    }

if(isNaN(step5))
    { //9901062327
QBspanAlert10.innerHTML=''
+'steps must be defined. Now it is ['
+QBboxSteps.value+']';
return;
    }

if(step5<2)step5=QBboxSteps.value=2;


	var MyGraph = new XYGraph();
	var MyLine1 = new XYLine();
	var MyLine2 = new XYLine();
	MyLine1.VMLpointshapetype="none";
	MyLine2.VMLpointshapetype="none";

status=''; 


	MyGraph.ymax=QBboxYmax10.value;
 MyGraph.ymin=QBboxYmin10.value;

	MyGraph.xmax=QBboxXmax10.value;
 MyGraph.xmin=QBboxXmin10.value;

	MyGraph.gheight=QBboxHeight10.value;
 MyGraph.gwidth=QBboxWidth10.value;

MyGraph.title=QBboxTitle10.value;

QBspanAlert10.innerHTML='';

QBdivGraph10.innerHTML = ""; 
QBspanEqnDoc10.innerHTML = "";

//above routine for all graph



if(rtArg1==21)
  { //9901171832

	MyGraph.ymax=+3;
 MyGraph.ymin=-2;

	MyGraph.xmax=+2;
 MyGraph.xmin=-3;

MyGraph.title='Exercise 6.14 complex roots in unit disk';

xbgn=0;
step0=60
delta=2*Math.PI/step0;

for(b0=0;b0<=step0;b0++)
      {
t=xbgn+delta*b0;
MyLine1.x[b0] =Math.cos(t)
MyLine1.y[b0] =Math.sin(t)
      } 
  MyLine1.VMLstroke="weight='1pt'; color='red'; dashstyle='solid';";
MyGraph.Plot(MyLine1); 

MyLine2.x[0] =-0.5
MyLine2.y[0] =-0.866025
MyLine2.x[1] =-2
MyLine2.y[1] = 0
MyLine2.x[2] =-0.5
MyLine2.y[2] = 0.866025
MyLine2.x[3] = 0.
MyLine2.y[3] = 0.
  MyLine2.VMLstroke="weight='1pt'; color='blue'; dashstyle='solid';";
MyGraph.Plot(MyLine2); 

	// draw intercept points
	var MyLabel = new Label();
	MyLabel.labelsize = "10";
//9506172108 try next
	MyLabel.VMLpointshapetype="circle";

MyLabel.labelcolor="black";

	MyLabel.x =0.2; MyLabel.y =0.8;
	MyLabel.label = 'A';
	MyGraph.Drawlabel(MyLabel);

	MyLabel.x =0.4; MyLabel.y =-0.6;
	MyLabel.label = 'B';
	MyGraph.Drawlabel(MyLabel);

	MyLabel.x =0.7; MyLabel.y =0.3;
	MyLabel.label = 'C';
	MyGraph.Drawlabel(MyLabel);

MyLabel.pointfillcolor="blue";

	MyLabel.x =-2; MyLabel.y =0.;
	MyLabel.label = 'z';
	MyGraph.Drawlabel(MyLabel);

	MyLabel.VMLpointshapetype="none";

	MyLabel.x =-1.8; MyLabel.y =-0.08;
	MyLabel.label = 'ψ';
	MyGraph.Drawlabel(MyLabel);


QBdivGraph21.innerHTML = MyGraph; 

return;
  } //9901171841


var delta=(xend-xbgn)/step5;
var x;

var eqn00=eqn01=eqn02=eqn03='';
eqn00=QBboxfunc0x.value; //9901061928
eqn01=QBboxfunc1x.value;
eqn02=QBboxfunc2x.value;
eqn03=QBboxfunc3x.value; //9901071046


if(rtArg1>50)
  { //9901071051

var t; //9901071053

if(eqn00.length>0&&eqn01.length>0)
    {
for(b0=0;b0<=step5;b0++)
      { //9901071054
t=xbgn+delta*b0;
MyLine1.x[b0] =eval('with(Math){'+eqn00+'}');
MyLine1.y[b0] =eval('with(Math){'+eqn01+'}');
      } //for(b0=0;b0<=step5;b0++)
  MyLine1.VMLstroke="weight='1pt'; color='red'; dashstyle='solid';";
MyGraph.Plot(MyLine1); 
    }

if(eqn02.length>0&&eqn03.length>0)
    {
for(b0=0;b0<=step5;b0++)
      { //9901071055
t=xbgn+delta*b0;
MyLine1.x[b0] =eval('with(Math){'+eqn02+'}');
MyLine1.y[b0] =eval('with(Math){'+eqn03+'}');
      } //for(b0=0;b0<=step5;b0++)
  MyLine1.VMLstroke="weight='1pt'; color='blue'; dashstyle='solid';";
MyGraph.Plot(MyLine1); 
    }

QBdivGraph10.innerHTML = MyGraph; 


QBspanEqnDoc10.innerHTML = ""
+"2010-01-07-11-09 create code<br>"
+"No document for this drawing.<br>"

return;
  } //if(rtArg1>50)




if(eqn00.length>0) //9901090805 use if()
  {
for(b0=0;b0<=step5;b0++)
    { //9812131606
x=xbgn+delta*b0; //9812131609

MyLine1.x[b0] =x;
MyLine1.y[b0] =eval('with(Math){'+eqn00+'}'); //9901061931

    } //for(b0=0;b0<=step5;b0++)

  MyLine1.VMLstroke="weight='1pt'; color='red'; dashstyle='solid';";
MyGraph.Plot(MyLine1); 
  } //if(eqn00.length>0)

if(eqn01.length>0)
  {
for(b0=0;b0<=step5;b0++)
    { //9812131638
x=xbgn+delta*b0; //9812131609

MyLine1.x[b0] =x;
MyLine1.y[b0] =eval('with(Math){'+eqn01+'}'); //9901061932
    } //for(b0=0;b0<=step5;b0++)

  MyLine1.VMLstroke="weight='1pt'; color='black'; dashstyle='solid';";
MyGraph.Plot(MyLine1); 
  } //if(eqn01.length>0)


if(eqn02.length>0)
  {
for(b0=0;b0<=step5;b0++)
    { //9812131643
x=xbgn+delta*b0;

MyLine1.x[b0] =x;
MyLine1.y[b0] =eval('with(Math){'+eqn02+'}'); //9901061933
    } //for(b0=0;b0<=step5;b0++)

  MyLine1.VMLstroke="weight='1pt'; color='blue'; dashstyle='solid';";
MyGraph.Plot(MyLine1); 
  } //if(eqn02.length>0)


if(eqn03.length>0)
  {
for(b0=0;b0<=step5;b0++)
    { //9901071048
x=xbgn+delta*b0;

MyLine1.x[b0] =x;
MyLine1.y[b0] =eval('with(Math){'+eqn03+'}'); //9901071049
    } //for(b0=0;b0<=step5;b0++)

  MyLine1.VMLstroke="weight='1pt'; color='purple'; dashstyle='solid';";
MyGraph.Plot(MyLine1); 
  } //if(eqn03.length>0)




if(rtArg1==11)
  {

QBspanEqnDoc10.innerHTML = ""
+"www.cms.math.ca/cjm/v28/cjm1976v28.0793-0804.pdf<br>"
+"page 794 (2/12) Lemma equation <br>"
+"p=lambda*y/[(1-lambda)*x+lambda*y]<br>"
+"above 'x' change to code fpar01<br>"
+"above 'y' change to code fpar02<br>"
+"above 'lambda' change to code x<br>"
+"fpar01 from p1 box, "+fpar01+"<br>"
+"fpar02 from p2 box, "+fpar02+"<br>"
+"Red curve: "+QBboxfunc0x.value+"<br>"
+"Black curve: "+QBboxfunc1x.value+"<br>"
+"Curve are defined for x in [0,1]<br>"
+"x outside of [0,1] may cause 1/0<br>"
+"2010-01-25-16-10<br>"

  }
else
if(rtArg1==12)
  {

QBspanEqnDoc10.innerHTML = ""
+"2010-01-24-15-00 start<br>"
+"How to create a Javascript code to draw above red<br>"
+"curve? <font color=red>Alert red curve has -infinite slope at x=0<br>"
+"Function pow(x,power) not allow x be negative.</font><br>"
+"2010-01-22-19-47 LiuHH access<br>"
+"http://marauder.millersville.edu/~bikenaga/calculus/conc/conc.pdf<br>"
+"found f0(x)=9*pow(x,4/3)/4-9*pow(x,1/3) example<br>"
+"Red curve    f0(x)=9*pow(x,4/3)/4-9*pow(x,1/3)<br>"
+"Black curve f'0(x)=3*pow(abs(x),1/3)-3*pow(abs(x),-2/3)<br>"
+"Blue curve f''0(x)=pow(abs(x),-2/3)+2*pow(abs(x),-5/3)<br>"
+"Write Javascript code, pow(x,1/3) change to pow(abs(x),1/3)<br>"
+"Use gatef(x,-7,0) to draw -x part<br>"
+"Use gatef(x,0,7) to draw +x part<br>"
+"<br>"
+"cubic root of -27 is -3 ( -3 cube is -27)<br>"
+"<font color=red>pow() use abs() and take away '-', we must make up '-'</font><br>"
+"When draw -x part, ODD fraction reverse +/- sign<br>"
+"For example f'0(x) is the sum of next two parts<br>"
+"gatef(x,-7,0)*(<font color=red size=+3><b>-3</b></font>*pow(abs(x),1/3)-3*pow(abs(x),-2/3))<br>"
+"+gatef(x,0,7)*(3*pow(abs(x),1/3)-3*pow(abs(x),-2/3))<br>"
+"Odd number '1' and '3' in '1/3' make 3 reverse to -3<br>"
+"Under 'gatef(x,-7,0)', abs() take away '-', reverse put '-' back.<br>"
+"Under 'gatef(x,0,7)', do not worry. Positive x, no trouble.<br>"
+"For better illustration, use abs(), did not replace by -x / +x<br>"
+"<br>"
+"f''0(x) is the sum of next two parts<br>"
+"gatef(x,-7,0)*(pow(abs(x),-2/3)<font color=red size=+3><b>-2</b></font>*pow(abs(x),-5/3))<br>"
+"+gatef(x,0,7)*(pow(abs(x),-2/3)+2*pow(abs(x),-5/3))<br>"
+"Odd number '5' and '3' in '-5/3' make 2 reverse to -2<br>"
+"<br>"
+"2010-01-24-15-12 record<br>"

  }

else
if(rtArg1==13)
  { //9901181940
QBdivGraph13.innerHTML = MyGraph; 


QBspanEqnDoc13.innerHTML = ""
+"2010-01-18-19-42 start<br>"
+"Textbook exercise 6.16, page 104/252<br>"
+"Red curve is f(x)=1/x<br>"
+"Black curve f'(x)=-1/x/x<br>"
+"Blue curve is f''(x)=2/x/x/x<br>"
+"Red curve is convex. Because blue<br>"
+"curve f''(x) > 0 for x > 0<br>"
+"2010-01-18-19-46 stop<br>"

return;

  }

QBdivGraph10.innerHTML = MyGraph; 

return;

} //function generalDrawf() //9812131648



//9901072310 include isAlpha(ch) to tute0026.htm
function isAlpha(ch) //9809151101
{
if((ch>='a'&&ch<='z')
 ||(ch>='A'&&ch<='Z')
  ) return true;
return false;
}




// 9507121343
function stepf(t0,bgn0)
{
  if(t0<bgn0) return 0.;
  return 1.;

} // end function // 9507121346
// 9507121356
// stepf(t,3) get t<3  value 0
// stepf(t,3) get t>=3 value 1
//
// (1-stepf(t,3)) get t<3  value 1
// (1-stepf(t,3)) get t>=3 value 0
//
// (stepf(t,2)-stepf(t,3))
//  get  2<=t<3  value 1
//  t<2 and t>=3 value 0
// 9507121400





//9804281114 gate function
function gatef(t0,bgn0,end0)
{
if(arguments.length==0)return 0;
if(arguments.length==1)return 0;
if(arguments.length==2) //9804281141
{ if(t0<bgn0) return 0;else return 1;}

  if(t0<bgn0) return 0.;
  if(t0>end0) return 0.;
  return 1.; // 9804281117
} // end function
// gatef(t,3.2,5.6) if t<3.2 get 0
// gatef(t,3.2,5.6) if t>5.6 get 0
// gatef(t,3.2,5.6) if 3.2<=t<=5.6 get 1
// 9804281119





/*
95,07,08,14,55,00
http://www.univie.ac.at/future.media/moe/JavaCalc/jcintro.html
c:\$fm\js\math9507\JavaCalc-jcintro.html

95,07,08,15,00
http://www.univie.ac.at/future.media/moe/JavaCalc/parser.js
c:\$fm\js\math9507\JavaCalc-jcintro-parser.js


Return to MathCollections

Ken Kikuchi
Comment me: kikuchi@mix.or.jp
Last updated: 2/29/2000

9507132210 include
function loggamma(x)
and
function gamma(x) 

/**/


// Ab hier (17. 3. 2000) von Ken's Script ernommen:

function factorial(n) {  /* factorial */
  with(Math) {
  if (n<0)  /* if negative */
  	return gamma(n+1);
  else if ((n == 0) || (n == 1))
    return 1;
  else if (abs(n)-floor(abs(n))==0 ) /* if positive integer */
    return n * factorial(n-1) ;
  else         /* if non-integer */
    return gamma(n+1);
} }


function loggamma(x)  { /* log gamma */
var u0; //9507142008

    with(Math) {
        var v=1;
        var w=0;
        var z=0;
        while ( x<8 ) { v*=x; x++ }
        w=1/(x*x);

u0= //9507142009
((((((((-3617/122400)*w + 7/1092)*w
         -691/360360)*w + 5/5940)*w
         -1/1680)*w + 1/1260)*w
         -1/360)*w + 1/12)/x 
+ 0.5 * log(2*PI)-log(v)-x+(x-0.5)*log(x) ;

//status='x=['+x+']; w=['+w+']; u0=['+u0+'] 9507142010';

        return u0;

} }


function gamma(x) {  /* gamma */

//9507141940
var i0;
var y0='';
var g0,g1,g2; //9507142019

for(i0=0;i0<(x+'').length;i0++)
{
  if(i0==0&&(x+'').charAt(i0)=='(')continue;
  if((x+'').charAt(i0)==')')break; //9507141944
  y0+=(x+'').charAt(i0);
}

x=parseFloat(y0);

    with(Math) {
        if ( x <= 0 )
        {
            if (abs(x)-floor(abs(x))==0 )
                return "ComplexInfinity" ;
            else
            {
g0=loggamma(1-x);
g1=PI/( sin(PI*x) * exp( g0 ) );
 return g1;
            }
        }
        else 
        {
g0=loggamma(x);
g1=exp( g0 ) ;
            return g1 ;
        }
} }


//ISBN 0-12-059820-5 page 307 line -4 
// LiuHH 200904271706
//function binomialCoef(arg1,arg2)
//function binoCoef(arg1,arg2)
function bicof(arg1,arg2)
{ //9804271706
arg1=parseInt(arg1);
arg2=parseInt(arg2);
if(arg1<arg2)return 0;
return factorial(arg1)/(factorial(arg2)*factorial(arg1-arg2));
} //9804271718


//function factorialDouble(arg1)
//ISBN 0-12-059820-5 page 545 eqn. 10.33c
// LiuHH 200904271746
//function factorialDouble(arg1)
//function factorDbl(arg1)
//
// Double factorial facdb(arg1)
// 9!!=9*7*5*3*1
// 8!!=8*6*4*2
function facdb(arg1)
{ //9804271746
arg1=parseInt(arg1);
var i0,i1,i2;
var n;

if(arg1%2)
  {
n=parseInt(arg1/2);
for(i0=0,i1=1;i0<n;i0++)
i1=i1*2;
return factorial(arg1)/factorial(n)/i1;
  } //9804271752
n=arg1/2;
for(i0=0,i1=1;i0<n;i0++)
i1=i1*2;
return factorial(n)*i1;
} //9804271754


//2010-01-06-19-56 inclide gamma function to tute0026.htm
//Main purpose is to add gatef() function to tute0026.htm


/**
2010-01-29-22-22 include
function bye09(in09)
to integral.htm


2009-06-12-15-31 start
Javascript use float number for calculation.
Answer may contain long string of '000000'
or '999999', for example
2.00000000000001 
-25.999999999999996 
On 2009-06-12-11-58 write a function
function bye09(in09) //9806121158
To use this function, call as following
coef2=bye09(coef2);
where coef2 is a number. After call
2.00000000000001 change to 2
-25.999999999999996 change to -26
It is handy, you can use it too.
If number is 1.23111111111 there is no change.

function bye09(in09) handle output number
for better looking. Do not call bye09()
during calculation, that is just slow down
process. If expect answer to be irrational
Do not call bye09().
If expect answer to be integer or short
decimal number like 1.2. In these case
you can call bye09().
2009-06-12-15-40 stop

2009-06-17-19-05
bye09() process only one number at a time.
If you have a string of several numbers.
Do not put number string as input argument
bye09() will send string back immediately.
This file has sample code at time stamp
'9806171858'
2009-06-17-19-09 stop

/**/
function bye09(in09) //9806121158
{ // in09 is input number,
// for example 2.0000000001
// If input is not a number, return here
if(isNaN(in09)) return in09;

// where is decimal point '.' ?
var dotLoc=(in09+'').indexOf('.',0);
// no decimal point? integer? return.
if(dotLoc==-1) return in09;

// change input string to number
in09=parseFloat(in09);

// if number has '000000' ？
var zeroLoc=(in09+'').indexOf('000000',dotLoc);
// find '000000' ，cut start '000000' 
if(zeroLoc>0) //9806121216
return parseFloat((in09+'').substring(0,zeroLoc)); //9806121209

// if number has '999999' ？
var nineLoc=(in09+'').indexOf('999999',dotLoc);
// find '999999' ， cut start '999999'
if(nineLoc>0) //9806121223 here
  { // copy number left to '999999'
//but immediately neighbor digit not copy
// nineLoc-1 because '999999' need increase
// digit by one.
//If input is 1.23999999999 copy only 1.2
//not copy 1.23 , because 3 will change to 4.
var nine0=(in09+'').substring(0,nineLoc-1);
//between '.' and '999999' has other digit.
if(nineLoc-dotLoc>1) //9806121220
    { //neighbor digit add one, paste to end
nine0+=(parseInt((in09+'').charAt(nineLoc-1))+1);
    }
else //between '.' and '999999' no other digit.
    {// modified number is an integer
nine0=parseInt(nine0); //9806121313
if(nine0<0)nine0--; // negative integer minus one
else nine0++; //9806121314 positive integer add one
    }
return parseFloat(nine0);
//9806121213 return new number
  }
// no '000000' no '999999' return original form
return parseFloat(in09); //9806121224
} //function bye09(in09) 


//-->
</script>

<br>

<hr>

<a name="differentiate">&lt;a name="differentiate"&gt;</a>
<br>
2010-02-09-22-32 differentiation　(<a href="#NumIntegral">Integration</a>)
<br>
f(x) is necessary, d[f(x)]/dx is arbitrary.
<br>
f(x)
  <INPUT id=QEboxf0
   value="exp(x)" size=50
  >

  <input onclick="javascript:QEboxf0.value=''"
 type="button" value="d" />

<br>
d[f(x)]/dx
  <INPUT id=QEboxf1
   value="exp(x)" size=42
  >

  <input onclick="javascript:QEboxf1.value=''"
 type="button" value="d" />

<br>

  <input onclick="javascript:QEspanDiffAns.innerHTML=dif1Man()"
 type="button" value="Differentiate" />

 x0 
  <INPUT id=QEboxX0
   value="1.2" size=8
  >
 ; dx
  <INPUT id=QEboxdx
   value="0.00001" size=8
  >

  <input onclick="javascript:QEboxX0.value=QEboxdx.value=''"
 type="button" value="d" />
 , 
  <input onclick="QEspanDiffAns.innerHTML='<br>'"
 type="button" value="dA" />

<br>
<font face='Courier New'>
<span id=QEspanDiffAns><br></span>
</font>
<br>
2010-02-09-22-39


<input onclick="javascript:
QEboxf0.value='exp(x)';
QEboxf1.value='exp(x)';
QEboxX0.value='1.2';
QEboxdx.value='1.e-6';
QEspanDiffAns.innerHTML=dif1Man()"
 type="button" value="51" /><!--9902101032-->

<input onclick="javascript:
QEboxf0.value='tan(x)';
QEboxf1.value='1/cos(x)/cos(x)';
QEboxX0.value='PI/2';
QEboxdx.value='-1.e-8';
QEspanDiffAns.innerHTML=dif1Man()"
 type="button" value="52" /><!--9902101033-->

<input onclick="javascript:
QEboxf0.value='pow((1+exp(x)),(-0.5))';
QEboxf1.value='-0.5*exp(x)*pow((1+exp(x)),(-1.5))';
QEboxX0.value='log(2)';
QEboxdx.value='1.e-6';
QEspanDiffAns.innerHTML=dif1Man()"
 type="button" value="53" /><!--9902101035-->

<input onclick="javascript:
QEboxf0.value='log(1+exp(x))';
QEboxf1.value='exp(x)/(1+exp(x))';
QEboxX0.value='1.2';
QEboxdx.value='1.e-6';
QEspanDiffAns.innerHTML=dif1Man()"
 type="button" value="54" /><!--9902101047-->

<input onclick="javascript:
QEboxf0.value='log(cos(x))';
QEboxf1.value='-sin(x)/cos(x)';
QEboxX0.value='1.2';
QEboxdx.value='1.e-6';
QEspanDiffAns.innerHTML=dif1Man()"
 type="button" value="55" /><!--9902101048-->


<br>


<script lang="javascript">
<!--

//9902100941
// differentiatef1() manager
//
function dif1Man()
{
var f0=QEboxf0.value;
var f1=QEboxf1.value;
var x0=QEboxX0.value;
var dx=QEboxdx.value;

x0=eval('with(Math){'+x0+'}'); //9902101159
dx=parseFloat(dx);

if(f0.length==0)
return 'need define function equation.<br>'
if(isNaN(x0))
return 'x0 must be a number.<br>'
if(isNaN(dx))
return 'increment dx must be a number.<br>'

var ans0=differentiatef1(f0,x0,dx); //9902100945
var fdx0=''; // df/dx answer
var x;
if(f1.length>0)
  {
x=x0;
fdx0=eval('with(Math){'+f1+'}')+'';
  }

return      "Numerical Answer : "+ans0+"<br>"
+(f1.length?"Analytic Solution: "+fdx0:"")
+"";

} //9902101004






/**
99,02,09,22,09
eqn00='sin(x)*cos(x)' = function definition
x0   =0.55 = evaluate point x0
dx   =0.0001 = delta_x
fx0  =20.2 = f(x0)

program evaluate second function
value at f(x0+dx)
use two function values  f(x0+dx)-f(x0)
use two domain values     x0+dx - x0
to find differentiate value at x=x0
d[f(x)]/dx = [f(x0+dx)-f(x0)] / dx
99,02,09,22,15 
/**/
function differentiatef1(dfArg1,dfArg2,dfArg3,dfArg4)
{
var eqn00=dfArg1;
var x0=parseFloat(dfArg2);
var dx=parseFloat(dfArg3);
var fx0=parseFloat(dfArg4);

if(eqn00.length==0)
return 'need define function equation.'
if(isNaN(x0))
return 'x0 must be a number.'
if(isNaN(dx))
return 'increment dx must be a number.'
if(arguments.length>3&&isNaN(fx0))
return 'f(x0) must be a number.'
var x=x0+dx;
var fx1=eval('with(Math){'+eqn00+'}')
if(arguments.length<=3)
  {
x=x0;
var fx0=eval('with(Math){'+eqn00+'}')
  }
return (fx1-fx0)/dx; //9902092228
}

//-->
</script>

<hr>


<a name="JSMathList">&lt;a name="JSMathList"&gt;</a>
<br>
<font color=red><b>
Equation definition must use Javascript language
</b></font>
<br>

&#9679;&#9679;Javascript math function
<INPUT name=FuncButton type="button"
 value="definition"
 onclick=javascript:open0lxr(101)> <!--9506291021-->

<br>

<!--ta ble+2a-->
<TABLE cellSpacing=1 cellPadding=0
 width="100%" align=center border=0>
<TBODY>
<TR>
</TR>
<TR id="showlxr101" style="DISPLAY: none"
 bgColor=#ccfcfc>
<TD colSpan=7>
<!--ta ble+3a-->
<TABLE width="100%" align=left border=0>
<TBODY>
<TR>
<TD align=right bgColor=#FFFF00 border=0
 onclick=open0lxr(101)>
C
<br>L
<br>O
<br>S
<br>E
</TD>
<TD width="100%">
<!--ta ble+4-->
<table >
  <tr>
  <td height="0%">
<pre><font size=3>
2010-01-30-11-32 add Javascript
math functions definition to
integral.htm

95,06,13,23,44
Javascript support the following math functions
[[
Math.E 2.718281828459045091 Euler's constant
Math.LN2 0.6931471805599452862 Natural log of 2
Math.LN10 2.302585092994045901 Natural log of 10
Math.LOG2E 1.442695040888963387 Log base-2 of E
Math.LOG10E 0.4342944819032518167 Log base-10 of E
Math.PI 3.141592653589793116  PI
Math.SQRT1_2 0.7071067811865475 Square root of 0.5
Math.SQRT2 1.414213562373095145 Square root of 2
Math.random() method takes no parameters
Math.abs(val) Absolute value of val
Math.acos(val) Arc cosine (in radians) of val
Math.asin(val) Arc sine (in radians) of val
Math.atan(val) Arc tangent (in radians) of val
Math.atan2(val1, val2) Angle of polar coordinates
Math.ceil(val) Next integer greater than or equal
Math.cos(val) Cosine of val
Math.exp(val) Euler's constant to the power of val
Math.floor(val) Next integer less than or equal to
Math.log(val) Natural logarithm (base e) of val
Math.max(val1, val2) The greater of val1 or val2
Math.min(val1, val2) The lesser of val1 or val2
Math.pow(val1, val2) Val1 to the val2 power
Math.random() Random number between 0 and 1
Math.round(val) N+1 when val >= n.5; otherwise N
Math.sin(val) Sine (in radians) of val
Math.sqrt(val) Square root of val
Math.tan(val) Tangent (in radians) of val
]]

for 
hyperbolic cosine, use
0.5*(Math.exp(x)+Math.exp(-x))

hyperbolic sine, use
0.5*(Math.exp(x)-Math.exp(-x))


Step function: stepf(a,b)
If a&lt;b return 0 (close)
otherwise return 1 (open)
Any number multiply by 0 get 0
that is to close.
Any number multiply by 1 get itself
that is to open.
Above is to multiply stepf()
If add or subtract stepf() then it
is shift curve up/down.
One of a,b must be variable. If
both a and b are constant, then
stepf() has no meaning. 9507211227


Gamma function  gamma(t)
If t is an integer gamma(t) is (t-1)!
gamma(t) let factorial become
continuous.
gamma(t) is searched from Internet.
9507211232

// gatef(t,3.2,5.6) if t<3.2 get 0
// gatef(t,3.2,5.6) if t>5.6 get 0
// gatef(t,3.2,5.6) if 3.2<=t<=5.6 get 1
// 9804281119

<hr>

2006-06-23-12-41 start
This program is received on 2006-06-12

2006-06-12-18-14<a href=http://www.bigwebmaster.com/3051.html>
http://www.bigwebmaster.com/3051.html</a>

2006-06-12-18-21<a href=http://www.structura.info/XYGraph/XYGraph.zip>
http://www.structura.info/XYGraph/XYGraph.zip</a>

=====

2006-10-23-09-08 start
Start from 2006-10-17 write this program.
Its URL is<a href=http://freeman2.com/graph09e.htm>
http://freeman2.com/graph09e.htm</a>
This file has same source as previous<a href=http://freeman2.com/graph03e.htm>
http://freeman2.com/graph03e.htm</a>
Both from XYGraph v2.3<a href=http://www.structura.info/XYGraph/XYGraphDemo.htm>
http://www.structura.info/XYGraph/XYGraphDemo.htm</a>
Freeman take their source code write a
better version.
graph03e.htm is mainly introduction
        All graph use one drawing board.
graph09e.htm is for application. Each
        graph has its own drawing board.
"has its own drawing board" is Freeman's
need. This function allow user to draw
multiple graph in one web page. Now
upload this program to Internet let
everyone use. Please click "Draw 606"
to see how to use.

graph09e.htm can dynamically create input
form and also one more change

graph03e.htm must use "Math.cos(t)"
          Where "Math." can not drop.
graph09e.htm allow to use cos(t)
          "Math." can be dropped.
Because graph09e.htm add 
[[
9510222211 use 'with(Math){' and '}'
]]
graph03e.htm follow XYGraph v2.3
             must use "Math."
graph09e.htm modified by Freeman
             no need to use "Math."

graph09e.htm still has limitation.
Curve allow maximum 5.
Arrow allow maximum 1.
Label allow maximum 1.
Equation can not be modified
Especially equation constant and 
number of terms.

graph03e.htm allow variable constant
             'a', 'b' to have five
             different values. (one
             equation family five
             members)
graph09e.htm broader this property to
             five arbitrary functions.

If you need function exceed above 
limitation, you must write code for it.
Examples can be found at<a href=http://freeman2.com/graph03e.htm>
http://freeman2.com/graph03e.htm</a>
find function Calculate0(PlotID){.....}
five examples.

Expect graph09e.htm will be used in<a href=http://freeman2.com/tutc0004.htm>
http://freeman2.com/tutc0004.htm</a>
This is Freeman's tutor page in Chinese.

Welcome visit Freeman site often.
Freeman 2006-10-24-18-01
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To open Javascript math function [definition]
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need external javascript file. If you can not
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click open, use a text editor open integral.htm
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search for the string
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"2010-01-30-11-32 add Javascript"
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and read from source file.
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2010-01-30-11-35

<pre><font size=+2>
<a name="docB001">&lt;a name="docB001"&gt;</a>
2010-01-27-00-01 create integral.htm

2010-01-27-00-08 start
Applied Numerical Methods, by
Brice Carnahan, H.A. Luther,
James O. Wilkes
1969 by John Wiley & Sons, Inc.
SBN 471-13507-0
(It is SBN, NOT ISBN)

<a name="docB002">&lt;a name="docB002"&gt;</a>
Page 73
Numerical Integration formula
equal space, close ends

n=1 (trapezoidal Rule), h=(x1-x0)/1
Integral[x=x0,x1]f(x)dx=
(h/2)*[f(x0)+f(x1)] - h^3*f''(ξ)/12

<a name="docB003">&lt;a name="docB003"&gt;</a>
Page 74
Numerical Integration formula
equal space, close ends

n=2 (Simpson's Rule), h=(x2-x0)/2
Integral[x=x0,x2]f(x)dx=
(h/3)*[f(x0)+4*f(x1)+f(x2)]
 - h^5*f<sup>(4)</sup>(ξ)/90

<a name="docB004">&lt;a name="docB004"&gt;</a>
n=3 (Simpson's Second Rule), h=(x3-x0)/3
Integral[x=x0,x3]f(x)dx=
(3*h/8)*[f(x0)+3*f(x1)+3*f(x2)+f(x3)]
 - h^5*f<sup>(4)</sup>(ξ)*3/80

n=4, h=(x4-x0)/4
Integral[x=x0,x4]f(x)dx=
(2*h/45)*[7*f(x0)+32*f(x1)+12*f(x2)
        +32*f(x3)+ 7*f(x4)]
 - h^7*f<sup>(6)</sup>(ξ)*8/945

<a name="docB005">&lt;a name="docB005"&gt;</a>
n=5, h=(x5-x0)/5
Integral[x=x0,x5]f(x)dx=
(5*h/288)*[19*f(x0)+75*f(x1)+50*f(x2)
        +50*f(x3)+75*f(x4)+19*f(x5)]
 - h^7*f<sup>(6)</sup>(ξ)*275/12096

<a name="docB006">&lt;a name="docB006"&gt;</a>
n=6, h=(x6-x0)/6
Integral[x=x0,x6]f(x)dx=
(h/140)*[41*f(x0)+216*f(x1)+27*f(x2)
        +272*f(x3)+27*f(x4)+216*f(x5)
        +41*f(x6)]
 - h^9*f<sup>(8)</sup>(ξ)*9/1400
2010-01-27-00-29 stop

<a name="docB007">&lt;a name="docB007"&gt;</a>
2010-01-30-09-53 start
This file 
http://freeman2.com/integral.htm
use Newton-Cotes formula n=1 to 6
calculate integration value for a
given function f(x) [box use g(x)]
Please goto <a href="#IntegralBgn">integration panel</a>
<a name="docB008">&lt;a name="docB008"&gt;</a>
It has the following arrangement
[[
Example: 1 2 3 4 5 6    debug 
g(x)  [ x*x*x-2*x*x+7*x-5 ]
 x bgn [1]  end [3]  steps  [5]
Newton-Cotes formula n=1 , 2, 3, 4, 5, 6 
message two lines
]]
<a name="docB009">&lt;a name="docB009"&gt;</a>
User need input function definition
at g(x) box. Program use 'x' as 
independent variable, then 'sin(x)'
will work, but 'sin(t)' do not work.
Integration need domain bounds,
x begin and x end. For example,
<a name="docB010">&lt;a name="docB010"&gt;</a>
if need integral[x=0,x=PI]{sin(x)dx}
input as following
[[
g(x)  [ sin(x) ]
 x bgn [0]  end [PI]  steps  [5]
]]
<a name="docB011">&lt;a name="docB011"&gt;</a>
Uppercase 'PI' and 'E' are defined
PI = 3.141592653589793
 E = 2.718281828459045
x bgn and end allow mathematics
expression. Next input
 x bgn [x=1;sin(x)]  end [vv=1.2;exp(vv)] 
program will automatically convert to
 x bgn =0.8414709848078965 
 x end =3.3201169227365472

<a name="docB012">&lt;a name="docB012"&gt;</a>
Next is about steps [ ] box.
Newton-Cotes formula coefficients
from n=1 to 6 are defined in program.
Assume choose n=3, program use
3/8; 1,  3,  3,  1 ; -3/80
<a name="docB013">&lt;a name="docB013"&gt;</a>
for calculation
[[
n=3 (Simpson's Second Rule), h=(x3-x0)/3
Integral[x=x0,x3]f(x)dx=
(3*h/8)*[f(x0)+3*f(x1)+3*f(x2)+f(x3)]
 - h^5*f(4)(ξ)*3/80
]]
<a name="docB014">&lt;a name="docB014"&gt;</a>
Apply Simpson's Second Rule (n=3)
(Newton-Cotes Formula ncfchoice=3)
need four points, x=x0,x1,x2,x3
evaluate to f(x0),f(x1),f(x2),f(x3)
oneSum to f(x0)+3*f(x1)+3*f(x2)+f(x3)
one integration to
(3*h/8)*[f(x0)+3*f(x1)+3*f(x2)+f(x3)]

<a name="docB015">&lt;a name="docB015"&gt;</a>
"steps [ ] box" is better understand
as "sub-intervals [ ] box". Because 
"steps" is shorter. Choose "steps".<!--9902091641-->
If set steps [ ] box to one, whole 
domain do Newton-Cotes formula once.
If set steps [ ] box to ten (stps=10).
Whole domain cut to ten equal section
do Newton-Cotes formula in each. If
Newton-Cotes formula n=3 and steps=2,
two equal section each need 4 points.
Whole domain divide to ncfchoice*stps
3*2=6 equal length xdelta 
xdelta=(xend-xbgn)/(ncfchoice*stps)
Total function evaluation =
      ncfchoice*stps+1

<a name="docB016">&lt;a name="docB016"&gt;</a>
steps [ ] box must use a positive 
integer. no other limit. In most 
case set to 1 is enough. <!--9902091642 start-->Other's
program no steps choice, that is
same as set steps=1. Larger steps
number, increase computing time,
and increase answer accuracy. If
increase steps number, answer do
not change accuracy any more. 
That is reached best answer. Test
with known answer problem, you
will find how steps number help.<!--9902091649 stop-->

<a name="docB017">&lt;a name="docB017"&gt;</a>
In
[[
Newton-Cotes formula n=1 , 2, 3, 4, 5, 6 
]]
specify one number. 
'1' indicate use polynomial of degree 1
    to approximate integration value.
'2' indicate use polynomial of degree 2
    to approximate integration value.
up to '6' are the same.

<a name="docB018">&lt;a name="docB018"&gt;</a>
Click button is
 [integral g(x) A] //output answer, no debug
 [integral g(x) B] //output answer and debug
<!--9902091716 bgn-->User can set "steps" as large as
possible. But large step debug 
output too long. Program set
limit at steps=100 or below do
debug. If steps=101 or greater
and if click [integral g(x) B]
program pop-up alert window and
not return debug data. Simply
because computer can not afford 
a debug request with step number
of 1000+.<!--9902091720 end-->
2010-01-30-10-40 here

<a name="docB019">&lt;a name="docB019"&gt;</a>
There are six example button 1 2 3 4 5 6
[1] is int[x=1,3]{x*x*x-2*x*x+7*x-5}dx
    steps=4, Newton-Cotes formula n=2
    answer 20.666666666666664 is exact.
    steps=1 is enough to get exact ans.
    SBN 471-13507-0 page 74 example.
<a name="docB020">&lt;a name="docB020"&gt;</a>
[2] is int[x=0,PI]{sin(x)}dx
    steps=10, Newton-Cotes formula n=6
    answer 2 is exact.
    steps=1 is not enough to get exact ans.
<a name="docB021">&lt;a name="docB021"&gt;</a>
[3] is int[x=0,PI]{1/(1+x*x)}dx
    steps=1, Newton-Cotes formula n=6
    answer 3.3287981274701663 is far
    away from 2*atan(4)
    steps=1  get answer 3.3287981274701663
    steps=10 get answer 2.651632548447489
    steps=20 get answer 2.651635326208792
    steps=30 get answer 2.651635327335626
    steps=40 get answer 2.651635327336065
    exact ans 2*atan(4)=2.651635327336065
<a name="docB022">&lt;a name="docB022"&gt;</a>
    g(x)=1/(1+x*x) is an example that
    Newton-Cotes formula may not converge
    to real answer. g(x)=1/(1+x*x) is from 
    ISBN 0-471-02985-8 page 229 example.

<a name="docB023">&lt;a name="docB023"&gt;</a>
An Introduction to Numerical Analysis
by Professor Kendall E. Atkinson
John Wiley & Sons Inc. 1978
This is LiuHH's textbook and
Professor K. E. Atkinson is instructor

<a name="docB024">&lt;a name="docB024"&gt;</a>
[4] is int[x=0,1]{sin(x)*sqrt(1-x*x)}dx
    steps=1, Newton-Cotes formula n=6
    answer 0.30644618846846266
    ISBN 0-471-02985-8 page 268 line -2.
    d[sin(x)*sqrt(1-x*x)]/dx singular at x=1
<a name="docB025">&lt;a name="docB025"&gt;</a>
[5] is int[x=0,1]{2*sin(1-x*x)*x*x*sqrt(2-x*x)}dx
    steps=1, Newton-Cotes formula n=6
    answer 0.3117417395331833
    ISBN 0-471-02985-8 page 268 line -2.
    d[g(x)]/dx is OK at x=1
    [4] equation change of variable get [5]
<a name="docB026">&lt;a name="docB026"&gt;</a>
[6] is<!--was int[x=sin(1),exp(2)]{x}dx
    2010-02-02-23-30 change to
   --> int[x=1,E]{1/x}dx
    steps=2, Newton-Cotes formula n=6
    answer 1.0
    Main point is to illustrate that x 
    upper bound E and lower bound exp(0)
    can be mathematics expression.
<a name="docB026a">&lt;a name="docB026a"&gt;</a>
    Original goal is allow user input PI
    as bounds. The code structure allow
    general mathematics expression.<font color=red><b>
    int[x=1,E]{1/x}dx=1 is an important
    integration, Euler use it found the
    unknown number E=2.718281828459045</b></font>
<a name="docB026b">&lt;a name="docB026b"&gt;</a>
    Why use integral value=1 to define E
    Why not use integral value=2 or
    other value to define E?
    Because log function always hang a 
    tail. Only if use integral value=1<!--tutc0005.htm#a06b14-->
    we get a tail "/1". Who bother to
    write a "/1" after a equation?
    Euler determined E value, let us
    use E base log function and tail
    is invisible. You hide my tail !
    Thank you Euler!    <!--2010-02-03-00-12 change stop-->

<a name="docB027">&lt;a name="docB027"&gt;</a>
As usual:<font color=red><b>
Output may contain error, Please verify
first.
Program environment is MSIE 6.0, please
 use MSIE.</b></font>

Thank you for visiting Freeman's web site.
Hope you find integral.htm interesting and
useful.
Freeman (Liu,Hsinhan)
2010-01-30-11-15 stop

<a name="docB028">&lt;a name="docB028"&gt;</a>
2010-01-30-14-01 start
Added "g2f" button, copy g(x) box function
to f0(x) box. Also delete f1(x) f2(x) f3(x)
boxes values. User modify 
x min:  , x max:  ; y min:  , y max:
and
Curve start x/t begin  , Curve end x/t end 
then click [Draw f0(x) to f3(x)] button
to see g(x) curve.

<a name="docB029">&lt;a name="docB029"&gt;</a>
If you want to compare two curves, second
function must hand copy and paste to f1(x)
box. At most draw 4 curves f2(x) and f3(x)
2010-01-30-14-10 stop

2010-02-01-14-51 start
<a name="docB030">&lt;a name="docB030"&gt;</a>
2010-01-30-17-56 LiuHH accessed next page<a href=http://www.zweigmedia.com/RealWorld/integral/integral.html>
http://www.zweigmedia.com/RealWorld/integral/integral.html</a>
This page has adaptive quadrature method 
option. Which is better than Newton-Cotes
formula. You can save the source file and
keep program in your computer.
2010-02-01-14-56 here

<a name="docB031">&lt;a name="docB031"&gt;</a>
2010-01-30-18-32 LiuHH accessed next page<a href=http://www.csse.monash.edu.au/~lloyd/tildeAlgDS/Numerical/Integration/>
http://www.csse.monash.edu.au/~lloyd/tildeAlgDS/Numerical/Integration/</a>
This page has document and display codes.
Page end has input boxes for 
f(x)= [ ]x^4+ [ ]x^3+ [ ]x^2+ [ ]x^1+ [ ]
You can save the source file and
keep program in your computer.
2010-02-01-15-06 stop

<a name="docB032">&lt;a name="docB032"&gt;</a>
2010-02-06-14-07 start
"Update 2010-02-06" has following changes
1. Add <a href="#docB030">link</a>.
2. "x-bgn", "x-end", "x/t begin", "x/t end"
   four boxes accept "PI" or "E" or math
   expression like
   "a=0;b=2;c=1;cos(a)*pow(b,c)"
<a name="docB033">&lt;a name="docB033"&gt;</a>
3. add example [7] button
   example [7] integrate f(x)=exp(-x)
   From x=0 to x=1000 simulate infinity.
   Taylor series expansion for exp(x) 
   has x^infinity, then exp(-x) has
   x^(-infinity). Newton-Cotes formula
<a name="docB034">&lt;a name="docB034"&gt;</a>
   n=6 use x^6 to integrate exp(-x)
   need large number of steps (division)
   to get exact answer
   ∫[x=0,infinity]exp(-x)dx = 1
   Other integration method (other than
   Newton-Cotes) may do better job.
2010-02-06-14-28 stop

<a name="docB035">&lt;a name="docB035"&gt;</a>
2010-02-06-14-32 start
How to use power function?
for example x^5 (x*x*x*x*x)
Some author spend time write code
convert from x^5 to pow(x,5).
Javascript accept only pow(x,5).
pow(2.345,6.789)=325.76152441868
pow(negative,fraction) get NaN
pow(negative,integer) get number
<a name="docB036">&lt;a name="docB036"&gt;</a>
This file no conversion code.
User must write pow() function
directly. For base^power, write
pow(base, power)
Both base and power two variables
must contain a numerical value.
2010-02-06-14-37 stop

<a name="docB037">&lt;a name="docB037"&gt;</a>
2010-02-09-17-01 start
"Update 2010-02-09" make x-bgn 
and x-end <a href="#IntegralBgn">boxes</a> larger for 
longer formula. 
Add comments about '<a href="#docB015">steps</a>'
2010-02-09-17-06 stop

<a name="docB038">&lt;a name="docB038"&gt;</a>
2010-02-10-10-15 start
"Update 2010-02-09" add <a href="#differentiate">
differentiate</a> function.
2010-02-10-10-15 stop
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