Integral Calculator - Definite Integrals of Any Function, with Infinite Limits
Integral Calculator
Area under a curve, to about 10 correct digits
Infinite Limits
Type Infinity or -Infinity as a limit to integrate over an unbounded range, for example the area under a bell curve.
Error Estimate
Every result comes with an estimate of its numerical error and the work the method needed.
Natural Notation
Write the function the way you would on a calculator, with pi, e and the usual functions.
Private
Everything is calculated in your browser.
How Numerical Integration Works
Most functions have no neat formula for their integral, but the area under a curve can still be calculated to high accuracy by sampling the function at carefully chosen points and adding the results with weights. This calculator uses Gauss-Kronrod quadrature, the method behind many scientific libraries: it estimates the area on an interval twice with two related rules, and the difference between the two tells it how accurate the answer is.
Where the estimate is not good enough, the interval is split in two and the process repeats, so the work goes where the function is difficult. Infinite limits are mapped onto a finite range by a change of variable. The result is a numerical value, not a formula, and it can be wrong when the function has a singularity inside the interval, jumps, or oscillates wildly, so check unusual cases another way. The integral of a function below the x-axis counts as negative area.
Key Takeaways
- Checked against mpmath: Results agree with high-precision reference values for a wide range of functions, including improper integrals.
- Signed area: Regions under the x-axis are subtracted. Swap the limits and the sign flips.
- Related calculators: See also the Derivative Calculator and the Graphing Calculator.