Solid of Revolution

To find the Volume of a solid of rotation by adding up a sequence of thin cylindrical shells, consider a region bounded above by $y = f(x)$, below by $y = g(x)$, on the left by the Line $x = a$, and on the right by the Line $x = b$. When the region is rotated about the y-Axis, the resulting Volume is given by

$$V = 2\pi \int_b^a x[f(x)-g(x)]\,dx.$$

To find the volume of a solid of rotation by adding up a sequence of thin flat disks, consider a region bounded above by $y = f(x)$, below by $y = g(x)$, on the left by the Line $x = a$, and on the right by the Line $x = b$. When the region is rotated about the x-Axis, the resulting Volume is

$$V = \pi\int_b^a \{[f(x)]^2-[g(x)]^2\}\,dx.$$

See also Surface of Revolution, Volume