Solid of Revolution (Torus)

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

|\x- 1/|2 + y2 = 0.25,
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

hi = /_\ yi,
inner radius
V~ ----------- r = 1- 0.25- |\y*/|2 i i
and outer radius
V~ ---------- R = 1+ 0.25- |\y*/|2 i i

The volume of ith washer is

V~ ----------- p\|R 2- r2/|h = 4p 0.25 - |\y*/|2 /_\ y i i i i i
as shown in Figure 2.

Figure 2

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

n V~ ----------- sum 4p 0.25 - |\y*/|2 /_\ y . i i i=1

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

integral 0.5 4p V~ 0.25---y2dy = p-2. -0.5 2

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

ri = x*i ,
thickness
wi = /_\ xi
and height
V~ --------------- h = 2 0.25 - |\x*- 1/|2. i i

The volume of the ith cylindrical shell is

V~ --------------- 2pr hw = 4p x* 0.25 - |\x*- 1/|2 /_\ x , i i i i i i
as shown in Figure 3.

Figure 3

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

n V~ --------------- sum * * 2 4pxi 0.25- |\xi - 1/| /_\ xi. i=1

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

integral 1.5 V ~ ------------2 p-2 0.5 4px 0.25- |\x - 1/| dx = 2 .