Introduction For this program, let\'s explore some of the ways we can use C to p

Question

Introduction For this program, let's explore some of the ways we can use C to perform numerical integration. We'll restrict our attention to definite integrals on smooth curves only and look at two methods, called the rectangle rule and the trapezoidal rule respectively With the rectangle rule, if we wish to calculate the area under a curve between two end points,a and b, we can evaluate the function at the midpoint between a and b and use that value to create a rectangle whose area will approximate the area under the curve between a and b. A diagram follows: f((a +b) 2) f(a) b x (a +b)/2 Here, the area of the red rectangle is consider an approximation for the definite integral between a and b. This area can be expressed by the following equation: [1(r) dz ~ (b-a) ,() When this technique is applied to a larger region of the curve, we can divide the whole large region into smaller sub-regions of equal size, and apply the rectangle rule to each one. We may get something like the following

Explanation / Answer

Given below is the code for the question. Please don't forget to rate the answer if it helped. Thank you.

#include <stdio.h>

typedef double (*mainFunction) (double);
typedef double (*calcArea) (mainFunction, double, double);

double curve1( double x ); //function f(x) for the curve
double calcAreaRect (mainFunction f, double a, double b); //function to calculate area using rectangle rule
double calcAreaTrap (mainFunction f, double a, double b); //function to calculate area using trapezoidal rule
double calcIntegral (mainFunction f, calcArea getArea, double a, double b); //calculte definite integral in interval a,b

int main()
{
int choice = 0;
double a, b, value = 0;
printf("Program to calculate the definite integral of curve f(x) = x^2 + 3x + 2 in the interval [a,b] ");
  
while(choice != 3)
{
printf("1. Rectangle method ");
printf("2. Trapezoidal method ");
printf("3. Exit ");
printf("Enter your choice: ");
scanf("%d", &choice);
  
if(choice != 3)
{
printf("Enter the interval [a,b] - ");
printf("a = ");
scanf("%lf", &a);
printf("b = ");
scanf("%lf", &b);
}
  
switch(choice)
{
case 1:
value = calcIntegral(curve1, calcAreaRect, a, b); //pass the function names as parameter
printf("Using Rectangle method, the definite integral for f(x) in the interval [%.2f, %.2f] is %.2f ", a, b, value);
break;
case 2:
value = calcIntegral(curve1, calcAreaTrap, a, b); //pass the function names as parameter
printf("Using Trapezoidal method, the definite integral for f(x) in the interval [%.2f, %.2f] is %.2f ", a, b, value);
case 3:
break;
default:
printf("Invalid choice!");
  
}
}
}
//function f(x) for the curve
double curve1( double x )
{
//f(x) = x^2 + 3x + 2
return x * x + 3 * x + 2;
}

//function to calculate area using rectangle rule
double calcAreaRect (mainFunction f, double a, double b)
{
return (b - a) * f((a+b)/2);
}
//function to calculate area using trapezoidal rule
double calcAreaTrap (mainFunction f, double a, double b)
{
return (b - a) * ((f(a) + f(b))/2);
}

//calculte definite integral in interval a,b
double calcIntegral (mainFunction f, calcArea getArea, double a, double b)
{
  
double a1, b1;
double area = 0;
int NUM_INTERVALS = 100;
double increment = (b - a) / NUM_INTERVALS;
  
a1 = a;
b1 = a + increment;
  
while(b1 <= b)
{
area = area + getArea(f, a1, b1);
a1 = b1;
b1 += increment;
}
  
return area;
  
}

output

Program to calculate the definite integral of curve f(x) = x^2 + 3x + 2 in the interval [a,b]
1. Rectangle method
2. Trapezoidal method
3. Exit
Enter your choice: 1
Enter the interval [a,b] -
a = 0
b = 1.0
Using Rectangle method, the definite integral for f(x) in the interval [0.00, 1.00] is 3.77
1. Rectangle method
2. Trapezoidal method
3. Exit
Enter your choice: 2
Enter the interval [a,b] -
a = 0
b = 1
Using Trapezoidal method, the definite integral for f(x) in the interval [0.00, 1.00] is 3.77
1. Rectangle method
2. Trapezoidal method
3. Exit
Enter your choice: 1
Enter the interval [a,b] -
a = 1
b = 3.0
Using Rectangle method, the definite integral for f(x) in the interval [1.00, 3.00] is 24.27
1. Rectangle method
2. Trapezoidal method
3. Exit
Enter your choice: 2
Enter the interval [a,b] -
a = 1.0
b = 3.0
Using Trapezoidal method, the definite integral for f(x) in the interval [1.00, 3.00] is 24.27
1. Rectangle method
2. Trapezoidal method
3. Exit
Enter your choice: 1
Enter the interval [a,b] -
a = 0.0
b = 3.0
Using Rectangle method, the definite integral for f(x) in the interval [0.00, 3.00] is 28.50
1. Rectangle method
2. Trapezoidal method
3. Exit
Enter your choice: 2
Enter the interval [a,b] -
a = 0.0
b = 3.0
Using Trapezoidal method, the definite integral for f(x) in the interval [0.00, 3.00] is 28.50
1. Rectangle method
2. Trapezoidal method
3. Exit
Enter your choice: 3

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