The idea

Volumes of revolution

Rotating a region about the x- or y-axis to make a solid, and finding its volume with V = π∫y² dx or V = π∫x² dy — squaring before integrating, matching the limits to the axis of rotation, and subtracting one volume from another.

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Pure · Integration

Volumes of revolution

Why it works

Slicing the solid into discs

Spin the region under a curve about an axis and it sweeps out a solid. To find its volume, slice the solid into thin discs perpendicular to the axis.

Rotating about the $x$ -axis, a slice at position

x

of thickness

\delta x

is a disc whose radius is the height of the curve,

y

. Its volume is

\pi y^2\,\delta x

. Adding the slices and letting

\delta x \to 0

V = \pi\int_a^b y^2\,\mathrm{d}x.

Rotating about the $y$ -axis, the discs are horizontal, the radius is now the horizontal distance

x

, and the thickness is

\delta y

V = \pi\int_c^d x^2\,\mathrm{d}y.

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