Solids of revolution

In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis) that lies on the same plane.

Assuming that the figure lies entirely on one side of the axis, the solid's volume is equal to the length of the circle described by the figure's barycenter, times the figure's area.

See also: surface of revolution

Formulas for solids of revolution

Rotations about the y-axis

The volume of the solid formed by rotating the area between the curves of <math>f(x)<math> and <math>g(x)<math> and the lines <math>x=a<math> and <math>x=b<math> about the y-axis is given by

<math>V = 2\pi \int_a^b x[f(x) - g(x)] dx<math>

If one of the bounding curves is actually the x-axis, then we can let <math>g(x) = 0<math> in the formula above, and we have:

<math>V = 2\pi \int_a^b x f(x) dx<math>

Rotations about the x-axis

The volume of the solid formed by rotating the area between the curves of <math>f(x)<math> and <math>g(x)<math> and the lines <math>x=a<math> and <math>x=b<math> about the x-axis is given by

<math>V = \pi \int_a^b [f(x)]^2 - [g(x)]^2 dx<math>

As above, we can use

<math>V = \pi \int_a^b [f(x)]^2 dx<math>

if one of the bounding curves is actually the x-axis.