Question

I need to solve math problem by using freemat or mathlab program. However, I can't understand how to make a code it.
This is one example for code ( freemat )

format long

f = @(x) (x^2) % define f(x) = x^2
a=0
b=1
N=23
dx = (b-a)/N % dx = delta_x

1. Methods:

(a) Left Rectangular Rule also known as Lower sum.

(b) Right Rectangular Rule also know as upper sum.

(c) Midpoint Rule

(d) Trapezoid Rule

(e) Simpson Rule

2. Evaluate

Integral {4/(x^2 + 1)} (Upper bound 1 Lower bound 0)

Answer 1

1. (a) Left Rectangular Rule : The left rectangular rule is the sum given by

for the partition

  

where

  

In case of uniform partition size we have

Create a MATLAB function file called Left_Rectangular_Rule.m and paste the code given below.

function L = Left_Rectangular_Rule(f,a,b,n)
%LEFT_RECTANGULAR_RULE Approximate the integral value using Left Rectangle
% Rule
% Input:
% f - Function handle of the integrand f(x)
% a - Lower limit of integeration
% b - Upper limit of integeration
% n - Number of partitions of interval [a,b]
% Output:
% L - Left Rectangle Rule Approximation of integral

% Calculate the step size delta x
dx = (b-a)/n;

% Calculate the Left Rectangle sum of the partition of [a,b]
L = dx*sum(f(a:dx:b-dx));

end

(b) Right Rectangular Rule : The Right Rectangular Rule is given by the sum

  

for the partition

  

where

  

In case of uniform partition size we have

Create a MATLAB function file called Right_Rectangular_Rule.m and paste the code given below.

function R = Right_Rectangular_Rule(f,a,b,n)
%Right_RECTANGULAR_RULE Approximate the integral value using Right Rectangle
% Rule
% Input:
% f - Function handle of the integrand f(x)
% a - Lower limit of integeration
% b - Upper limit of integeration
% n - Number of partitions of interval [a,b]
% Output:
% R - Right Rectangle Rule Approximation of integral

% Calculate the step size delta x
dx = (b-a)/n;

% Calculate the Right Rectangle sum of the partition of [a,b]
R = dx*sum(f(a+dx:dx:b));

end

(c)  Midpoint Rule : The mid point rule is the sum given by

  

where

are the mid points of the partitions of the interval

where

  

In case of uniform partition size we have

Create a MATLAB function file called MidPoint_Rule.m and paste the code given below.

function M = MidPoint_Rule(f,a,b,n)
%MIDPOINT_RULE Approximate the integral value using Mid point
% Rule
% Input:
% f - Function handle of the integrand f(x)
% a - Lower limit of integeration
% b - Upper limit of integeration
% n - Number of partitions of interval [a,b]
% Output:
% M - Mid Point Rule Approximation of integral

% Calculate the step size delta x
dx = (b-a)/n;

% Calculate the mid points of the partitions of [a,b]
m = ((a:dx:b-dx) + (a+dx:dx:b))/2;

% Calculate the Mid Point Rule sum of the partition of [a,b]
M = dx*sum(f(m));

end

(d) Trapezoidal Rule : The trapezoidal rule is given by the sum

for the partition

  

where

  

In case of uniform partition size we have

Create a MATLAB function file called Trapezoidal_Rule.m and paste the code given below.

function T = Trapezoidal_Rule(f,a,b,n)
%TRAPEZOIDAL_RULE Approximate the integral value using Trapezoidal Rule
% Input:
% f - Function handle of the integrand f(x)
% a - Lower limit of integeration
% b - Upper limit of integeration
% n - Number of partitions of interval [a,b]
% Output:
% M - Trapezoidal Rule Approximation of integral

% Calculate the step size delta x
dx = (b-a)/n;

% Calculate the Trapezoidal Rule sum of the partition of [a,b]
T = (dx/2)*sum(f(a:dx:b-dx)+ f(a+dx:dx:b));

end

2. We can now approximate the integral value

  

using the 4 functions we have defined.

The results of the approximations are given below.

The exact value of the integral is