Calculate Integral C++

What is an Integral?

An integral represents the area under a curve between two points. In calculus, it's the reverse of differentiation. For functions that can't be integrated analytically, numerical methods provide approximate solutions.

Numerical integration approximates this area using computational methods. Common techniques include the trapezoidal rule, Simpson's rule, and Monte Carlo integration.

Numerical Integration Methods

Trapezoidal Rule

Divides the area under the curve into trapezoids and sums their areas. More intervals provide better accuracy.

Simpson's Rule

Uses parabolas instead of straight lines, providing higher accuracy with fewer intervals.

Monte Carlo Integration

Uses random sampling to estimate the integral. More samples improve accuracy.

C++ Implementation

Here's a C++ implementation of the trapezoidal rule for numerical integration:

#include <iostream>
#include <cmath>

double trapezoidalRule(double (*func)(double), double a, double b, int n) {
    double h = (b - a) / n;
    double sum = (func(a) + func(b)) / 2.0;

    for (int i = 1; i < n; i++) {
        sum += func(a + i * h);
    }

    return sum * h;
}

double myFunction(double x) {
    return sin(x);
}

int main() {
    double a = 0.0;
    double b = M_PI;
    int n = 1000;

    double result = trapezoidalRule(myFunction, a, b, n);
    std::cout << "Integral result: " << result << std::endl;

    return 0;
}

The code defines a function to integrate, implements the trapezoidal rule, and calculates the integral of sin(x) from 0 to π.

Example Calculation

Let's calculate the integral of f(x) = x² from 0 to 1 using the trapezoidal rule with 10 intervals.

The exact value is 1/3 ≈ 0.3333. Using the trapezoidal rule with 10 intervals gives approximately 0.3337, which is very close to the exact value.