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Definite Integral Calculator

Work out definite integral instantly with clear inputs, formula shown and shareable results.

About Definite Integral Calculator

The fundamental theorem of calculus evaluates a definite integral as F(b) − F(a), where F is any antiderivative. Simpson's rule is exact for cubics, so it confirms the result here.

Formula

Fundamental theorem

∫ₐᵇ f(x) dx = F(b) − F(a)

Worked example

With the default values below, Definite Integral Calculator returns Definite integral of 54.00000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Polynomial coefficients, highest power first
3, 0, -2, 1
Lower limit a
1
Upper limit b
3

Results

Definite integral
54.00000000
Simpson's rule check
54.00000000
F(b)
54.75000000
F(a)
0.75000000

Step by step

  1. 1Values used
  2. 2Fundamental theorem
  3. 3Definite integral= 54.00000000
  4. 4Simpson's rule check= 54.00000000
  5. 5F(b)= 54.75000000
  6. 6F(a)= 0.75000000

Assumptions and limits

  • The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Integrate 3x³ − 2x + 1 from 1 to 3.

F(x) = 0.75x⁴ − x² + x, giving F(3) − F(1) = 54.75 − 0.75 = 54.

Can a definite integral be negative?

Yes. It is a signed area, so regions below the axis count negatively.

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