Solid of Revolution

A solid of revolution is a solid enclosing the surface of revolution obtained by rotating a 1-dimensional curve, line, etc. about an axis. A portion of a solid of revolution obtained by cutting via a plane oblique to its base is called an ungula.

To find the volume of a solid of revolution by adding up a sequence of thin cylindrical shells, consider a region bounded above by , below by , on the left by the line , and on the right by the line . When the region is rotated about the z-axis, the resulting volume is given by

The following table gives the volumes of various solids of revolution computed using the method of cylinders.

solid			volume
cone		0
conical frustum		0
cylinder		0
oblate spheroid
prolate spheroid
sphere
torus
spherical segment
torispherical dome		0

To find the volume of a solid of revolution by adding up a sequence of thin flat washers, consider a region bounded on the left by , on the right by , on the bottom by the line , and on the top by the line . When the region is rotated about the z-axis, the resulting volume is

The following table gives the volumes of various solids of revolution computed using the method of washers.

solid			volume
barrel (elliptical)		0
barrel (parabolic)		0
cone		0
conical frustum		0
cylinder		0
oblate spheroid		0
prolate spheroid		0
sphere		0
torus
spherical segment		0
torispherical dome		0