Calculus 2nd Edition Rogawski

334 CHAPTER 6 APPLICATIONS OF THE INTEGRAL
4. Compressing from equilibrium to 4 cm past equilibrium
5. Stretching from 5 cm to 15 cm past equilibrium
6. Compressing 4 cm more when it is already compressed 5 cm
7. If 5 J of work are needed to stretch a spring 10 cm beyond equilibrium,
how much work is required to stretch it 15 cm beyond equilibrium?
8. To create images of samples at the molecular level, atomic force
microscopes use silicon micro-cantilevers that obey Hooke’s Law
F (x) = −kx, where x is the distance through which the tip is deflected
(Figure 6). Suppose that 10 −17 J of work are required to deflect the tip
a distance 10 −8 m. Find the deflection if a force of 10 −9 N is applied
to the tip.
16. Calculate the work (against gravity) required to build a box of
height 3 m and square base of side 2 m out of material of variable density,
assuming that the density at height y is f (y) = 1000 − 100y kg/m 3 .
In Exercises 17–22, calculate the work (in joules) required to pump all
of the water out of a full tank. Distances are in meters, and the density
of water is 1000 kg/m 3 .
17. Rectangular tank in Figure 8; water exits from a small hole at the
top.
Water exits here.
1
5
Laser beam
Surface
Tip
Cantilever
8
FIGURE 8
4
18. Rectangular tank in Figure 8; water exits through the spout.
10000 nm
19. Hemisphere in Figure 9; water exits through the spout.
FIGURE 6
9. A spring obeys a force law F (x) = −kx 1.1 with k = 100 N/m.
Find the work required to stretch a spring 0.3 m past equilibrium.
2
10
10. Show that the work required to stretch a spring from position
a to position b is 1 2 k(b2 − a 2 ), where k is the spring constant.
How do you interpret the negative work obtained when |b| < |a|?
In Exercises 11–14, use the method of Examples 2 and 3 to calculate the
work against gravity required to build the structure out of a lightweight
material of density 600 kg/m 3 .
11. Box of height 3 m and square base of side 2 m
12. Cylindrical column of height 4 m and radius 0.8 m
13. Right circular cone of height 4 m and base of radius 1.2 m
14. Hemisphere of radius 0.8 m
15. Built around 2600 bce, the Great Pyramid of Giza in Egypt (Figure
7) is 146 m high and has a square base of side 230 m. Find the work
(against gravity) required to build the pyramid if the density of the stone
is estimated at 2000 kg/m 3 .
FIGURE 9
20. Conical tank in Figure 10; water exits through the spout.
2
5
FIGURE 10
21. Horizontal cylinder in Figure 11; water exits from a small hole at
the top. Hint: Evaluate the integral by interpreting part of it as the area
of a circle.
10
Water exits here.
r
FIGURE 7 The Great Pyramid in Giza, Egypt.
FIGURE 11