Definite Integral Calculator
Adaptive Simpson, with the error it actually has.
Work out Definite Integral. Adaptive Simpson, with the error it actually has. Shows the working, not just the answer.
Use * for multiplication and ^ for powers.
Definite integral
9
Adaptive Simpson's rule, 16 intervals · error below 1e-15
Adaptive Simpson's rule, doubling the interval count until two successive estimates agree. Simpson is exact for any cubic, so its error falls as the fourth power of the step — halving the step cuts the error sixteenfold, which is why it converges so quickly on smooth functions. The error figure is a Richardson estimate from that convergence rather than a guess, and it is shown because quoting twelve digits from a method good for eight is a lie told confidently. A singularity at either limit is refused rather than nudged inward. Shifting the endpoint by a whisker to avoid an infinity silently integrates a different problem and returns a plausible number for it.
How the Definite Integral Calculator works
The definite integral of a typed function between two limits, by adaptive Simpson's rule, with an honest error estimate. Quoting twelve digits from a method good for eight is a lie told confidently.
Also known as: area under a curve · definite integral solver · integrate between limits · numerical integration calculator
Frequently asked questions
What does a definite integral represent?
The signed area between the curve and the x-axis over the interval. Area below the axis counts as negative, which is why the integral of sin from 0 to 2π is zero rather than four.
How accurate is the result?
The error estimate is reported alongside. Simpson's rule is exact for any cubic and its error falls as the fourth power of the step size, so it converges very quickly on smooth functions — usually to machine precision within a few doublings.
Why does it refuse an integral with a singularity at a limit?
Because nudging the endpoint inward to avoid the infinity silently integrates a different problem and returns a plausible-looking number for it. Improper integrals need a limit process, not a fudge.
What is the average value row?
The integral divided by the width of the interval — the constant height that would enclose the same area. It is the mean value theorem for integrals, made concrete.
Can it do indefinite integrals?
No. Symbolic antidifferentiation is a far harder problem than differentiation and many elementary functions have no elementary antiderivative at all — exp(−x²) being the famous case. This handles the definite case, which is what most questions actually need.
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