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MATLAB – Trapezoidal numerical integration without using trapz

  • Difficulty Level : Medium
  • Last Updated : 04 Jul, 2021

Trapezoidal rule is utilized to discover the approximation of a definite integral. The main idea in the Trapezoidal rule is to accept the region under the graph of the given function to be a trapezoid rather than a rectangle shape and calculate its region.

The formula for numerical integration using trapezoidal rule is:

∫_a^bf(x)dx=h/2[ f(a)+2\{ ∑^{n-1}_{i=1}f(a+ih)\}  +f(b)]

where h = (b-a)/n

Now we take an example for calculating the area under the curve ∫_0^11/(1+x^2) dx     using 10 subintervals.



Example:

Matlab




% MATLAB program for calculate the 
% area under the curve ∫_0^11/(1+x^2) dx 
% using 10 subintervals specify the variable
% x as symbolic ones The syms function creates 
% a variable dynamically and automatically assigns
% to a MATLAB variable with the same name
syms x
  
% Lower Limit
a=0;  
  
% Upper Limit
b=1;   
  
% Number of segments
n=10; 
  
% Declare the function
f1=1/(1+x^2);
  
% inline creates a function of
% string containing in f1
f=inline(f1); 
  
% h is the segment size
h=(b - a)/n;
  
% X stores the summation of first
% and last segment
X=f(a)+f(b);
  
% variable R stores the summation of
% all the terms from 1 to n-1
R=0;
for i = 1:1:n-1
    xi=a+(i*h);
    R=R+f(xi);
end
  
% Formula to calculate numerical integration
% using Trapezoidal Rule
I=(h/2)*(X+2*R);
  
% Display the output
disp('Area under the curve 1/(1+x^2) = ');
disp(I);

Output:

Let’s take another example for calculating the area under the curve ∫_0^1x^2 dx      using 4 subintervals.

Example:

Matlab




% MATLAB program for calculate
% the area under the curve∫_0^1x^2 dx    
% using 4 subintervals.
% specify the variable x as symbolic ones
  
syms x
  
% Lower Limit
a=0; 
  
% Upper Limit
b=1; 
  
% Number of segments
n=4;  
  
% Declare the function
f1=x^2;
  
% inline creates a function of
% string containing in f1
f=inline(f1); 
  
% h is the segment size
h=(b - a)/n;
X=f(a)+f(b);
  
% variable R stores the summation 
% of all the terms from 1 to n-1
R=0;
for i = 1:1:n-1
    xi=a+(i*h);
    R=R+f(xi);
end
  
% Formula to calculate numerical
% integration using Trapezoidal Rule
I=(h/2)*(X+2*R);
  
% Display the output
disp('Area under the curve x^2 = ');
disp(I);

Output:




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