Question

Suppose that a function f(x) is given and the computation of its value for any x has been implemented by a function with prototype

double f(double x);

Write a function that computes the integral of f(x) over the range (a, b) for any input values a and b. The function is to be implemented using the Trapezoidal rule and must return a double typed value.

Answer 1

To calculate the integral of f(x) over [a,b], we can use the Trapezoidal rule, which approximates the area under the curve with trapezoids. The implementation involves dividing the interval into subintervals, summing weighted function values, and scaling by the subinterval width to approximate the integral.

Step-by-step explanation:

To compute the integral of a function f(x) over the range (a, b) using the Trapezoidal rule, we can implement a function in programming code. The Trapezoidal rule works by approximating the area under the curve as a series of trapezoids rather than rectangles. The function would take two input parameters, a and b, which define the interval over which to integrate, and would return a double-type value that represents the definite integral approximation.

Here is a pseudocode implementation:

function trapezoidalIntegral(double a, double b, int n) {
double h = (b - a) / n;
double sum = f(a) + f(b);
for (int i = 1; i < n; ++i) {
double x_i = a + i * h;
sum += 2 * f(x_i);
}
return (h / 2) * sum;
}

This code snippet divides the interval [a,b] into n equal subintervals, computes the sum of the values of f(x) at the endpoints and twice the sum of the values of f(x) at all interior points, and multiplies by half the width of the subintervals (h/2).