Numerical Integration Calculator

Approximate definite integrals using Trapezoidal and Simpson's rules. Visualize the area under the curve.

Function (e.g., x^2, sin(x), x*cos(x))

Lower Bound

Upper Bound

Method

Approximate Integral Value:

Visualization

Understanding Numerical Integration

Numerical integration is used to approximate the value of a definite integral when an analytical solution is difficult or impossible to find. This tool uses two common methods: the Trapezoidal Rule and Simpson's Rule.

Trapezoidal Rule

The Trapezoidal Rule approximates the area under the curve by dividing it into trapezoids. The formula is:

$$ \int_a^b f(x) \, dx \approx \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right] $$

where $ h = \frac{b-a}{n} $ and $ x_i = a + i \\cdot h $, and $ n $ is the number of trapezoids (intervals).

Simpson's Rule

Simpson's Rule uses quadratic polynomials to approximate the curve, generally providing a more accurate approximation than the Trapezoidal Rule, especially for smooth functions. The formula is:

$$ \int_a^b f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right] $$

where $ h = \frac{b-a}{n} $, $ n $ must be an even number, and $ x_i = a + i \\cdot h $.

Using the Calculator

Enter your function in terms of 'x' (e.g., sin(x), x^2 + 2x).
Specify the lower and upper bounds of integration.
Choose the numerical method: Trapezoidal or Simpson's Rule.
Click 'Calculate' to get the approximate integral value.
View the visualization of the function and the approximated area under the curve.