Trapezoidal Rule with MATLAB Program Example

Trapezoidal Rule Derivation

The derivation for obtaining formula for Trapezoidal rule is given by,

Example

Evaluate the integral x^4 within limits -3 to 3 using Trapezoidal rule.

Solution

Let y(x)=x^4

here a=-3 and b=3

therefore (b-a)=6

let ‘n’ be the number of intervals. assume n=6 in this case.

also h=(b-a)/n = 6/6 =1

x: -3  -2  -1  0  1  2  3

y: 81  16  1  0  1  16  81

According to trapezoidal rule:

answer= (h/2)*[(y1+y7)+2*(y2+y3+y4+y5+y6)]

answer=(1/2)*[(81+81)+2*(16+1+0+1+16)]

answer=115.

MATLAB Program for Trapezoidal Rule

%Created by myclassbook.org (Mayuresh)
 %Created on 24 May 2013
 %Question: Evaluate the integral X^4 within limits 3 to -3

clc;
 clear all;
 close all;

f=@(x)x^4; %Change here for different function
 a=-3;b=3; %Given limits
 n=b-a; %Number of intervals
 h=(b-a)/n;
 p=0;

for i=a:b
  p=p+1;
  x(p)=i;
  y(p)=i^4; %Change here for different function
 end

l=length(x);
 x
 y
 answer=(h/2)*((y(1)+y(l))+2*(sum(y)-y(1)-y(l)))

Image format

Example

Evaluate the integral 1/(1+x) within limits 0 to 6 using Trapazoidal rule.

Solution

Let y(x)=1/(1+x)

here a=0 and b=6

therefore (b-a)=6

let ‘n’ be the number of intervals. assume n=6 in this case.

also h=(b-a)/n = 6/6 =1

x: 0                  1                    2              3                  4               5               6

y: 1.0000   0.5000   0.3333   0.2500   0.2000   0.1667   0.1429

According to trapazoidal rule:

answer= (h/2)*[(y1+y7)+2*(y2+y3+y4+y5+y6)]

answer=2.0214.

MATLAB code for Trapazoidal rule :

%Created by myclassbook.org (Mayuresh)
%Created on 24 May 2013
%Question: Evaluate the integral 1/(1+x) within limits 0 to 6

clc;
 clear all;
 close all;

f=@(x)1/(1+x); %Change here for different function
 a=0;b=6; %Given limits
 n=b-a; %Number of intervals
 h=(b-a)/n;
 p=0;

for i=a:b
  p=p+1;
  x(p)=i;
  y(p)=1/(1+i); %Change here for different function
 end

l=length(x);
 x
 y
 answer=(h/2)*((y(1)+y(l))+2*(sum(y)-y(1)-y(l)))

Image Format

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