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Area when the graph changes side

The definite integral has a sign. Area above the xx-axis counts as positive, and area below the axis counts as negative. If you add both parts, they can cancel even though the geometric area is not zero. You then split at the zeros and take the absolute value of each part. For an odd function about the origin, the integral from −a-a to aa is zero, while the area is twice one side.

Try it yourself

The integral as area

∫−aax3 dx=0\int_{-a}^{a} x^3\,dx = 0with sign
2∫0a∣x3∣ dx2\int_0^a |x^3|\,dxthe geometric area

Symbols

aathe zero

Example

∫−11x3 dx=14−14=0\int_{-1}^{1} x^3\,dx = \dfrac{1}{4} - \dfrac{1}{4} = 0. The area between the graph and the axis is 2⋅14=0.52\cdot\dfrac{1}{4} = 0.5.

An integral equal to zero does not mean that the area is zero. Check whether the graph lies below the axis.
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Part of Mathematics R2: Area and volume with integrals.