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Definite Integral Calculator

Evaluate a definite integral numerically with adaptive Simpson's rule. See the value, the error estimate and a shaded plot of the area under the curve.

0.33333333
Integral value
0
Error estimate
8
Subintervals used

Result

∫ x^2 dx from 0 to 1 = 0.3333333333

Computed by adaptive composite Simpson's rule. The interval count doubles until the Richardson error estimate |S2n − Sn| / 15 drops below the chosen tolerance. This is a numerical answer, not a symbolic antiderivative.

Area plot

What is the Definite Integral Calculator?

The ByteTools Definite Integral Calculator evaluates ∫ₐᵇ f(x) dx using adaptive composite Simpson's rule.

  • Adaptive composite Simpson's rule with a Richardson error estimate
  • Reports the error bound and the number of subintervals used
  • Shaded area plot with the integration limits marked
  • Flags points inside the interval where the integrand breaks down
  • Handles reversed limits by flipping the sign, as the definition requires
  • 100% local — the function and limits never leave your browser

How to use the Definite Integral Calculator

  1. 1

    Type the integrand in the f(x) box, for example x^2, sin(x) or exp(-x^2).

  2. 2

    Enter the lower limit a and the upper limit b.

  3. 3

    Pick a tolerance — 1e-8 suits most work, tighter values take more subintervals.

  4. 4

    Read the value, the error estimate and the subinterval count, and check the shaded area plot.

  5. 5

    Click Copy result to save the value and its error bound.

About the Definite Integral Calculator

The ByteTools Definite Integral Calculator evaluates ∫ₐᵇ f(x) dx using adaptive composite Simpson's rule. It keeps doubling the number of subintervals until the Richardson error estimate |S₂ₙ − Sₙ|/15 drops under the tolerance you pick, then reports the value, that error estimate and how many subintervals it needed to get there.

The area between the curve and the x-axis is shaded on a plot so you can see what the number actually represents, with the limits marked. If the integrand has a point inside the interval where it has no finite value, the page flags it instead of returning a confident-looking answer built on a hole in the function.

This is a numerical method, not a symbolic antiderivative, and the page says so plainly. Everything is computed in your browser — no data leaves your device and the tool works offline.

Frequently asked questions

How accurate is a numerical definite integral?

For a smooth integrand, adaptive Simpson's rule is very accurate — often to ten or more digits. The page shows its own error estimate so you can judge. Accuracy drops sharply if the function has a spike, a kink or a vertical asymptote inside the interval.

What is the difference between a definite and an indefinite integral?

A definite integral has limits and evaluates to a single number, usually interpreted as signed area under the curve. An indefinite integral is the family of antiderivatives, written with a + C. This calculator computes the definite kind numerically.

Why is my integral negative?

Area below the x-axis counts as negative in a definite integral, so a curve that spends more time under the axis than above it gives a negative total. Swapping the limits also flips the sign, which the calculator handles and tells you about.

What happens if the limits are the same?

The integral is exactly zero by definition, because there is no interval to integrate over. The calculator returns 0 and says why, rather than dividing by a zero-width interval and producing nonsense.

Can this calculator do improper integrals to infinity?

No — it needs two finite limits. Simpson's rule samples the function at points across a bounded interval, so an infinite limit or an unbounded integrand falls outside what it can honestly compute.

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