What is Simpson's rule?

Simpson's rule is a numerical integration technique that approximates the area under a curve by fitting parabolic segments to subintervals. It is significantly more accurate than the trapezoidal rule for smooth functions.

What functions are supported?

You can use standard JavaScript math expressions: x^2 (or x**2), sin(x), cos(x), tan(x), Math.exp(x) or exp(x), log(x) for natural log, sqrt(x), and any combination with +, -, *, /.

How accurate is the result?

For smooth functions, Simpson's rule with 1000 subintervals typically gives accuracy to 6 or more significant figures. Increase the subinterval count for functions with sharp changes.

Why do I get NaN or Infinity?

This happens when the function has a singularity (like 1/x at x=0) or is undefined on the interval. Adjust your bounds to avoid discontinuities.

Can I compute improper integrals?

This tool supports finite closed intervals only. For improper integrals with infinite bounds, use a large but finite approximation (e.g. 0 to 1000 instead of 0 to infinity).

Area Under Curve Calculator

Enter any math function and interval to instantly compute the definite integral using Simpson's rule numerical integration.

Function & Interval

f(x) — Function

Use x as variable. Operators: +, -, *, /, ^ or **. Functions: sin, cos, tan, sqrt, exp, log, abs.

Lower bound (a)

Upper bound (b)

Subintervals (n) — must be even

Quick Examples

Enter a function and click Calculate to see the result.

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Summary

The Area Under Curve Calculator computes definite integrals numerically using Simpson's rule, a highly accurate numerical integration method. Enter a mathematical function, specify the lower and upper bounds, and the tool instantly approximates the area under the curve with adjustable precision. Ideal for students, engineers, and scientists who need quick integral estimates without symbolic computation.

How it works

Enter a mathematical function in the f(x) field (e.g. x^2, sin(x), e^x).
Set the lower bound (a) and upper bound (b) of the integration interval.
Choose the number of subintervals (higher = more accurate).
Click "Calculate" to run Simpson's rule numerical integration.
The result displays the approximate area and a summary of the method used.

Use cases

Estimate the area under a velocity-time graph to find displacement.
Approximate definite integrals for functions without closed-form antiderivatives.
Verify calculus homework answers with a quick numerical check.
Compute areas for irregular curves in engineering design.
Analyze probability distributions by integrating PDF functions.
Explore the effect of changing interval bounds on total area.

Frequently Asked Questions

Last updated: 2026-05-29 · Reviewed by Nham Vu