6.2. USING DEFINITE INTEGRALS TO FIND VOLUME
351
x
y
Figure 6.11: The solid of revolution described in Example 6.4.
by r(x) = x 2 + 1, while the outer radius is R(x) = x + 1. Therefore, the volume of a typical
slice is
V slice = π[R(x)
2 − r(x)
2 ]△x = π
(x + 1)
2 − (x
2 + 1)
2
△x.
Finally, we integrate to find the total volume, and
V =
1
0
π
(x + 1)
2 − (x
2 + 1)
2
dx =
7
15
π.
Activity 6.6.
In each of the following questions, draw a careful, labeled sketch of the region described,
as well as the resulting solid that results from revolving the region about the stated
axis. In addition, draw a representative slice and state the volume of that slice, along
with a definite integral whose value is the volume of the entire solid. It is not necessary
to evaluate the integrals you find. For each prompt, use the finite region S in the first
quadrant bounded by the curves y = 2x and y = x 3 .
(a) Revolve S about the line y = −2.
(b) Revolve S about the line y = 4.
(c) Revolve S about the line x = −1.
(d) Revolve S about the line x = 5.
⊳
Summary