# Simpson’s 3/8th Rule – C PROGRAM

Simpson’s 3/8th Rule is a Numerical technique to find the definite integral of a function within a given interval.
It’s so called because the value 3/8 appears in the formula.

The function is divided into many sub-intervals and each interval is approximated by a cubic curve. And the area is then calculated to find the integral. The more is the number of sub-intervals used, the better is the approximation.

### FORMULA:

$\int_a^b f(x)dx= \frac{3h}{8}(f_0 + 3f_1 + 3f_2 + 2f_3 + 3f_4 + 3f_5 + 2 f_6 ....+ 3f_{n-1} + f_n)$
where,
$f_i=a+ih$ where $i$ starts from 0 and goes to $n$
NOTE: The no. of sub-intervals $n$, should be a multiple of 3 for this method.

### PROGRAM:

/*********************************
*******SIMPSON'S 3/8 RULE********
********************************/
#include<stdio.h>
#include<math.h>
double f(double x){
return x*x;
}
main(){
int n,i;
double a,b,h,x,sum=0,integral;
printf("\nEnter the no. of sub-intervals(MULTIPLE OF 3): ");
scanf("%d",&n);
printf("\nEnter the initial limit: ");
scanf("%lf",&a);
printf("\nEnter the final limit: ");
scanf("%lf",&b);
h=fabs(b-a)/n;
for(i=1;i<n;i++){
x=a+i*h;
if(i%3==0){
sum=sum+2*f(x);
}
else{
sum=sum+3*f(x);
}
}
integral=(3*h/8)*(f(a)+f(b)+sum);
printf("\nThe integral is: %lf\n",integral);
}


### OUTPUT:

[wpedon id="7041" align="center"]

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