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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)
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
(Radians Mode)
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

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