Definite Integral Calculator
Evaluate definite integrals accurately with composite Simpson's rule, with control over the number of intervals.
Category: Mathematics
About this tool
What Is a definite integral? Definite Integral Calculator. The definite integral calculator evaluates ∫ₐᵇ f(x) dx using composite Simpson's rule, a high-order numerical method that approximates the area under the curve to many digits for smooth functions. A definite integral sums an infinite set of tiny slices to find the net area between f(x) and the x-axis from a to b. Negative slices count below the axis, so an integral can be zero even when the curve is not. Simpson's rule fits parabolas through triplets of points instead of straight lines, which gives error of order h⁴ — excellent accuracy for modest interval counts like 1000 used by default. ∫ₐᵇ f(x) dx via Simpson: (h/3)(f₀ + 4f₁ + 2f₂ + … + fₙ), n even. How to Use Enter the function in x, like x^2 or sin(x). Enter the lower limit a and upper limit b. Optionally change the interval count n (even, 2–20000); larger n means more precision. Read the result immediately. Worked Example Integrate x² from 0 to 1. The tool returns ≈ 0.33333333, matching the exact 1/3. Integrating sin(x) from 0 to π gives ≈ 2, the exact net area of one hump. Common Use Cases area, volume and average-value problems work and energy calculations in physics probability densities and means checking analytic integration homework Important Notes The rule needs an even interval count and the tool enforces that. If a > b the direction is swapped and the