Page 237 - Calculus Demystified

CHAPTER 8
224
Applications of the Integral
We easily evaluate this integral as follows:
3 1
x
V = π · x −
3
−1

1 −1
= π · 1 − − −1 −
3 3
4
= π.
3
You Try It: Any book of tables (see [CRC]) will tell you that the volume inside a
3
sphere of radius r is 4πr /3. This formula is consistent with the answer we obtained
in the last example for r = 1. Use the method of this section to derive this more
general formula for arbitrary r.

8.2 Volumes of Solids of Revolution

8.2.0 INTRODUCTION

Auseful way—and one that we encounter frequently in everyday life—for generat-
ing solids is by revolving a planar region about an axis. For example, we can think
of a ball (the interior of a sphere) as the solid obtained by rotating a disk about
an axis (Fig. 8.12). We can think of a cylinder as the solid obtained by rotating
a rectangle about an adjacent axis (Fig. 8.13). We can think of a tubular solid as
obtained by rotating a rectangle around a non-adjacent axis (Fig. 8.14).

Fig. 8.12

Fig. 8.13