The region bounded by the circle with center at (1, 0) and radius 1/2,
is revolved about the yaxis, generating the solid shown in Figure 1. This doughnutshaped solid is called a torus.




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 xaxis.
Partition the interval [0.5, 0.5] on the yaxis into n subintervals and construct horizontal rectangles to approximate the area of the circle. Each rectangle, when revolved about the yaxis, generates a washer. The ith washer has height
inner radius
and outer radius
The volume of ith washer is
as shown in 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
Partition the interval [0.5, 1.5] on the xaxis into n subintervals and construct vertical rectangles to approximate the area of the circle. The ith rectangle, when revolved about the yaxis, generates a cylindrical shell with radius
thickness
and height
The volume of the ith cylindrical shell is
as shown in 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