Average value
3
Math & numbers · Calculus
Integral
Integral
calculator
Enter a function and two limits, and get the definite integral — the signed area under the curve — along with the average value of f over the interval, computed live in the page.
∫ dxSimpson's rule · 2,000 slices
∫ x^2 dx from 0 to 3
9
Interval width
3
∫ₐᵇ f(x) dx ≈ (h/3) [ f₀ + 4f₁ + 2f₂ + … + fₙ ]
The area under any curve, to eight decimal places
A definite integral adds up a function's values across an interval — area under a curve, distance from a speed, total from a rate. Type the function and the two limits and this tool evaluates ∫ₐᵇ f(x) dx immediately, along with the average value of f over the interval, which is simply the integral divided by the width.
How it is computed
Simpson's rule · 2,000 slices
fits a parabola through each triple of points — exact for cubics
fits a parabola through each triple of points — exact for cubics
Simpson's rule approximates the curve piecewise with parabolas rather than rectangles, which makes it dramatically more accurate for the same effort: for smooth functions the error shrinks with the fourth power of the slice width. With two thousand slices, everyday integrals are correct to well past the eight decimal places shown — ∫ x² from 0 to 3 returns exactly 9.
Sign and direction
area below the axis counts negative
swapping the limits flips the sign: ∫ᵇₐ = −∫ₐᵇ
swapping the limits flips the sign: ∫ᵇₐ = −∫ₐᵇ
Two honest limits. First, this is a definite integral only — it returns a number, not an antiderivative expression. Second, the interval must be one the function actually lives on: integrating 1/x across 0, or tan(x) across π/2, spans a singularity, and no fixed slicing can rescue that — the tool reports a dash rather than a made-up value. For slopes instead of areas, the derivative calculator is the mirror tool, and the advanced calculator runs the same command in notebook form alongside solve and plot.
Common questions
Integral FAQ
No — it computes definite integrals only, returning the number ∫ₐᵇ f(x) dx rather than a formula. If you need the symbolic antiderivative you want a computer algebra system; if you need the value over an interval, this is the faster and often more practical tool.
For smooth functions, extremely: Simpson's error falls with the fourth power of the slice width, and it is exact for any polynomial up to degree three. Typical results are correct beyond the eight decimal places displayed. Rapidly oscillating functions over wide ranges are the main case that erodes that.
Area below the x-axis counts as negative, so the integral is a signed area. ∫ sin(x) dx from 0 to 2π is 0 because the hump above the axis exactly cancels the one below. If you want geometric area regardless of sign, integrate abs(f(x)) instead.
The sign flips: integrating from b back to a gives exactly the negative of integrating from a to b. The tool computes whatever direction you enter, so a larger first limit simply produces the negated value — that is the mathematics, not an error.
Not reliably. An interval that crosses a singularity — 1/x through zero, tan(x) through π/2 — or one that runs to infinity is outside what fixed slicing can do, and the tool shows a dash when the numbers blow up. Keep the interval inside the function's domain and it will behave.
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