Overview
A definite integral represents the net area between a function's curve and the x-axis over an interval. Positive values contribute positive area and negative values subtract from it. The calculator evaluates the integral numerically with Simpson's rule, which is very accurate for smooth functions, and also reports the average value over the interval.
Formula
∫ₐᵇ f(x) dx = F(b) − F(a), where F is an antiderivative of f.
How it works
Write the function
Enter the expression f(x) using x.
Set the integration limits
Enter a and b; a may be greater than b.
Choose the interval count
More intervals usually improve numerical accuracy.
Read the result
The integral, average value and endpoint values appear.
Worked examples
Example 1: ∫₀² x² dx
The result is 8 ÷ 3 ≈ 2.66666667, with average value 4 ÷ 3 ≈ 1.33333333.
Example 2: ∫₀^π sin x dx
The result is 2, the area above the axis.
Frequently asked questions
When is the integral negative?
When the area below the x-axis is greater than the area above it over the interval.
Can the lower limit be greater than the upper limit?
Yes, and the integral becomes the negative of the original value.
Why must the interval count be even?
Simpson's rule needs an even number of intervals, so the calculator automatically raises an odd count.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.