## Transcribed Text

Numerical Integration
Problems:
1. Write a user defined function named trapezoid that receives as input the
interval endpoints a, b, the value n, and the function f to be integrated.
Your funciton should use the Composite Trapezoidal Rule to return the
approximate value of the definite integral of the function.
2. Write a user defined function named simpson that receives as input
the interval endpoints a, b, the value n(even), and the function f to be
integrated. Your function should use the Composite Simpson’s Rule to
return the approximate value of the definite integral of the function.
3. Use your functions (trapezoid, simpson) to approximate the integral
∫ 1
0
e
x
2
dx.
Compare the values obtained for n = 2, 4, 6, 10. Use the error terms to
determine the smallest value of n needed to approximate the integral
to within 10
−2 and evaluate your functions with that value of n.
4. Determine the value of n required to approximate the integral
∫ 3
0
x
√
x
2 + 1
dx
to within 10
−5 accuracy with the Composite Simpson’s Rule. Compare
your actual error for this n to the theoretical error bound.
1
5. Use Romberg integration to compute R3,3 for
∫ 4
1
(
sin
2
(x) − 2x sin(x) + 1
)
dx
2
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function Solution()

clear all

clc

close all

disp('----------Question 3-------------')

%syms x

%Iexact=double(int(exp(x^2),x,0,1));

n=[2 4 6 10];

f=@(t)exp(t.^2);

for i=1:3

ErrorSimpson(i)=Simpson(f,0,1,n(i+1))-Simpson(f,0,1,n(i));

ErrorTrapezoid(i)=Trapezoid(f,0,1,n(i+1))-Trapezoid(f,0,1,n(i));

end

ErrorSimpson=ErrorSimpson

ErrorTrapezoid=ErrorTrapezoid

%determine the smallest i such that abs(ErrorTrapezoid(i))>10^(-2)

it=3; err=1;%initialize

while abs(err)>10^(-2)

err=Trapezoid(f,0,1,it)-Trapezoid(f,0,1,it-1);

it=it+1;

end...