## Saturday, August 24, 2019

### HP Prime and TI Nspire CX CAS: Solving Integral Equations

HP Prime and TI Nspire CX CAS:  Solving Integral Equations

Introduction

The program INTEGRALSOLVE solve the following equation:
(Format:  ∫( integrand dvar, lower, upper)

∫( f(t) dt, 0, x) = a

∫( f(t) dt, 0, x) - a = 0

It is assumed that x>0.

We can use the Second Theorem of Calculus which takes the derivative of the integral:

d/dx  ∫( f(t) dt, a, x) = f(x)

We don't have to worry about lower limit a at all for the theorem to work.

∫( f(t) dt, 0, x) - a

Take the derivative with respect to x on both sides (d/dx):

= d/dx ∫( f(t) dt, 0, x) - a

= d/dx ∫( f(t) dt, 0, x) - d/dx a

Let F(t) be the anti-derivative of f(t):

= d/dx (F(x) - F(0)) - 0

= d/dx F(x) -  d/dx F(0)

F(0) is a constant.

= f(x)

Newton's Method to find the roots of f(x) can be found by the iteration:

x_(n+1) = x_n - f(x_n) / f'(x_n)

Applying that to find the roots of ∫( f(t) dt, 0, x) - a:

x_(n+1) = x_n - (∫( f(t) dt, 0, x_n) - a) / f(x_n)

HP Prime Program INTEGRALSOLVE

Enter f(X) as a string as it will be stored in Function App variable F0.  Use X as the independent variable.

EXPORT INTEGRALSOLVE(f,a,x)
BEGIN
// f(X) as a string, area, guess
// ∫(f(X) dX,0,x) = a
// EWS 2019-07-26
// uses Function app
LOCAL x1,x2,s,i,w;
F0:=f;
s:=0;
x1:=x;
WHILE s==0 DO
i:=AREA(F0,0,x1)-a;
w:=F0(x1);
x2:=x1-i/w;
IF ABS(x1-x2)<1 font="" then="">
s:=1;
ELSE
x1:=x2;
END;
END;

RETURN approx(x2);
END;

TI NSpire CX CAS Program INTEGRALSOLVE
(Caution:  This program needs to be typed in)

Use t as the independent variable.

Define LibPub integralsolve(f,a,x)=
Func
:© f(x), area, guess: ∫(f(t) dt,0,x = a)
:Local x1,x2,s
:s:=0
:x1:=x
:While s=0
:  x2:=x1-((∫(f,t,0,x1)-a)/(f|t=x1))
:  If abs(x2-x1)≤1E−12 Then
:    s:=1
:  Else
:    x1:=x2
:  EndIf
:EndWhile
:Return approx(x2)
:EndFunc

Examples

Example 1:

∫( 2*t^3 dt, 0, x) = 16
Guess = 2
Root ≈ 2.3784

Example 2:

∫( sin^2 t dt, 0, x) = 1.4897
Guess = 1
Root ≈ 2.4999

Source:

Green, Larry.  "The Second Fundamental Theorem of Calculus"  Differential Calculus for Engineering and other Hard Sciences.  Lake Tahoe Community College. http://www.ltcconline.net/greenl/courses/105/Antiderivatives/SECFUND.HTM  Retrieved July 25, 2019

Happy Solving!

Eddie

All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

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