Solid of Revolution (Torus)

The region bounded by the circle with center at (1, 0) and radius 1/2,

is revolved about the y-axis, generating the solid shown in Figure 1. This doughnut-shaped solid is called a torus.

 Figure 1

The volume of this solid can be approximated by first approximating the area of the planar region with rectangles and revolving these rectangles about the x-axis.

Volume using washers

Partition the interval [-0.5, 0.5] on the y-axis into n subintervals and construct horizontal rectangles to approximate the area of the circle. Each rectangle, when revolved about the y-axis, generates a washer. The ith washer has height

The volume of ith washer is

as shown in Figure 2.

 Figure 2

The approximate volume of the solid is just the sum of the volumes of the circular disks,

The exact volume of the solid, of course, is given by the definite integral

Volume using cylindrical shells

Partition the interval [0.5, 1.5] on the x-axis into n subintervals and construct vertical rectangles to approximate the area of the circle. The ith rectangle, when revolved about the y-axis, generates a cylindrical shell with radius

thickness
and height

The volume of the ith cylindrical shell is

as shown in Figure 3.

 Figure 3

The approximate volume of the solid is just the sum of the volumes of the cylindrical shells,

The exact volume of the solid, of course, is given by the definite integral