Definite Integral Calculator

Find the area under the curve without the headache — give us a function and bounds, we'll crunch it.

∫ dx

from a = to b =

Uses Simpson's rule (numeric). Highly accurate for smooth functions.

What a definite integral tells you

A definite integral of f(x) from a to b measures the signed area between the curve and the x-axis across that interval. Area above the axis counts as positive, area below counts as negative, so a symmetric wave can integrate to zero even though it clearly encloses space. That single number answers a lot of practical questions: the distance covered when f(x) is a velocity, the total charge when f(x) is a current, the work done when f(x) is a force, or the expected value when f(x) is a probability density.

Why use a numerical calculator instead of solving by hand

Plenty of honest-looking functions have no elementary antiderivative — e^-x², sin(x)/x, and √(1+x³) among them. You can't write down a clean formula for their integral, but you can still get the number. This tool uses Simpson's rule, which fits parabolas across pairs of subintervals rather than flat rectangles, so it lands close to the true value with far fewer slices than a basic Riemann sum. For any smooth curve you'll usually get several correct digits almost instantly.

A quick worked example

Enter f(x) = x² with a = 0 and b = 3. The exact answer is 3³/3 = 9, and the calculator returns 9 because Simpson's rule is exact for polynomials up to cubics. Try f(x) = 1/x from 1 to 2 and you'll get about 0.6931, which is ln(2) — a value with no simple closed form.

How to use it

Type your function of x in the f(x) box. Use * for multiplication (3*x, not 3x), ^ for powers, and names like sin, cos, exp, ln, and sqrt.
Set the lower limit a and the upper limit b. You can enter b smaller than a; the sign of the result simply flips.
Optionally raise the number of subintervals for a sharp or wiggly curve, then read the computed area.

FAQ

Why did I get an error or a strange number?: Usually the interval crosses a point where the function blows up, like 1/x through 0, or the syntax is off — check for a missing * or an unmatched bracket.
How accurate is the result?: For smooth functions Simpson's rule is very close, often to 6+ digits. Sharp spikes or near-vertical sections need more subintervals to settle down.
Can it handle infinite limits?: Not directly. Replace infinity with a large finite bound (say 100) if the function decays quickly, and watch whether the value stabilizes as you push the bound further out.
Is it free and private?: Yes — it's free and ad-supported, and the whole calculation runs in your browser, so nothing you type is uploaded.