Definite Integral Calculator — Step-by-Step Solutions

f(x)

Lower Bound (a)

Upper Bound (b)

Your calculations will appear here

The Definite Integral Calculator evaluates integrals between specified upper and lower bounds. It uses numerical integration methods (Simpson's rule) to compute accurate results for a wide range of functions.

Enter your function, variable, lower bound, and upper bound to calculate the definite integral. The result represents the signed area under the curve between the specified limits.

How to Use the Definite Integral Calculator

Enter the function to integrate.
Specify the variable of integration and the lower and upper bounds.
The calculator uses Simpson's rule with 1000 intervals for high accuracy.

Formula and Theory

A definite integral computes the signed area between a function and the x-axis over a given interval [a, b]. Areas above the x-axis are positive and areas below are negative.

Simpson's rule is a numerical integration method that approximates the function using parabolic segments. It is significantly more accurate than the trapezoidal rule for smooth functions.

Worked Examples

Example 1: Polynomial integral

Problem: Evaluate the integral of x^2 from 0 to 3

Solution: Antiderivative: x^3/3. F(3) - F(0) = 27/3 - 0

Answer: 9

Example 2: Trigonometric integral

Problem: Evaluate the integral of sin(x) from 0 to pi

Solution: Antiderivative: -cos(x). -cos(pi) - (-cos(0)) = 1 + 1

Answer: 2

Frequently Asked Questions

What does the definite integral represent geometrically?

It represents the net signed area between the function and the x-axis over the interval. Area above the x-axis counts as positive, below as negative.

How accurate is numerical integration?

Simpson's rule with 1000 intervals typically gives results accurate to 8 or more decimal places for smooth functions. Accuracy may decrease for functions with discontinuities.

Can the bounds be reversed?

Yes. Reversing the bounds changes the sign: the integral from b to a = negative of the integral from a to b.