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Calculus basics

Definite integral (area under a curve)

The signed area between a curve and the x-axis from a to b, from any antiderivative F.

∫abf(x)dx=F(b)−F(a)

Calculator

The live calculator needs JavaScript. The worked example below shows the same steps it would.

See it move

−3−2−1123−4−224
  • Integral from −2 to 2 ≈ 10.6667 (shaded; parts below the axis count as negative).

What the symbols mean

I
the value of the integral: the signed area
c3
the coefficient of x3 in f
c2
the coefficient of x2
c1
the coefficient of x
c0
the constant term
a
the lower limit
b
the upper limit
F
an antiderivative of f (so F′=f)

When to use it

Use it for areas under curves, total distance from a speed graph, total change from a rate, and probabilities from density functions. The calculator integrates any cubic c3x3+c2x2+c1x+c0 exactly and checks the answer with Simpson's rule. Area below the axis counts as negative; for the total geometric area, split the interval where the curve crosses the axis.

Worked example

Find the area under y = 4 − x² between x = −2 and x = 2 (c₃ = 0, c₂ = −1, c₁ = 0, c₀ = 4, a = −2, b = 2).

  1. The function is

    f(x)=−x2+4

  2. Integrate term by term with the power rule (raise each power by one and divide by the new power):

    F(x)=−x33+4⁢x

  3. Evaluate at the upper limit:

    F(2)=−233+4·2
    =−83+8
    ≈5.33333

  4. Evaluate at the lower limit:

    F(−2)=−(−2)33+4·(−2)
    =−(−8)3+(−8)
    =83+(−8)
    ≈−5.33333

  5. Subtract: the integral is F(b) − F(a).

    I=5.33333−(−5.33333)
    ≈10.6667

  6. Numerical check: Simpson's rule with 200 strips gives 10.66666667, which agrees.

Answer I ≈ 10.667 square units

Why it works

This is the fundamental theorem of calculus: the area function A(x)=∫axf grows at rate f(x), so it is an antiderivative of f. Any two antiderivatives differ by a constant, which cancels in F(b)−F(a).

Common mistakes

  • Subtracting in the wrong order: it is F(upper) − F(lower).
  • Treating a negative integral as an area; the area is its absolute value.
  • Integrating straight across an x-intercept when the question asks for total area: the parts cancel.
  • Dropping brackets when substituting a negative limit: −(−2)3/3=+8/3.