Page 249 - Calculus Demystified

CHAPTER 8
236
EXAMPLE 8.16 Applications of the Integral
Imagine that a water tank in the shape of a hemisphere of radius 10 feet
isbeing pumped out (Fig. 8.26). Find the work done in lowering the water
level from 1 foot from the top of the tank to 3 feet from the top of the tank.

Radius at depth x
_ 2
equals √100 x

Fig. 8.26

SOLUTION
A glance at Fig. 8.27 shows that the horizontal planar slice of the tank, at the
√
2
level x feet below the top, is a disk of radius 100 − x . This disk therefore
2
has area A(x) = π · (100 − x ). Thus a slice at that level of thickness x will
have volume
2
V(x) = π · (100 − x ) · x

100
x
_ 2
√100 x

Fig. 8.27

and (assuming that water weights 62.4 pounds per cubic foot) weight equal to
2
w(x) = 62.4π · (100 − x ) · x.
Thus the work in raising this slice to the top of the tank (where it can then
be dumped) is

2
W(x) = 62.4π · (100 − x ) · x · x foot-pounds.