NAMES:
MATH 152
February 11, 2015
QUIZ 3
• Show all your work and indicate your final answer clearly. You will be graded not merely
on the final answer, but also on the work leading up to it.
2
1. (3 points) Find√the volume of the solid formed when the region bounded by y = ex , x = 0,
y = 0 and x = ln 2 is revolved around the y-axis.
Solution: The radius of a cylindrical shell will be x and the height of the shell will be
2
determined by the function ex . Thus
√
Zln 2
2
V = 2π
xex dx.
0
2
2
du
2
Letting u = x imples du = 2x dx ⇒
= x dx. So the antiderivative of xex is
Z
Z
1 2
1
1
x2
eu du = eu = ex
xe dx =
2
2
2
and
√
Zln 2
2
V = 2π
xex dx
0
√ln 2
= πe x2
0
ln 2
= π(e
− e0 )
= π(2 − 1) = π.
2. (3 points) A uniform cable hanging over the edge of a tall building is 40 feet long and weighs
60 pounds. How much work is required to pull 10 feet of the cable to the top?
Solution: The top of the building is the origin with down the positive y direction. The cable
weighs 1.5 lb/ft. A slice of thickness dy in the top 10 feet of cable weighs 1.5 · dy lb and is
lifted y feet. Thus, the work on a slice in the top 10 feet of the cable is Wslice = 23 y dy ft-lb.
A slice of the remaining 30 feet of cable weighs 23 · y lb and is lifted 10 feet so the work on
a slice of this part of the rope is Wslice = 32 · 10 ft-lb. Thus the total work is
40
10
Z 10
Z 40
3
3
3 2 W =
x dx +
· 10 dx = x + 15x = 525 ft-lb.
2
4 0
0
10 2
10
NAMES:
MATH 152
February 11, 2015
3. (3 points) An aquarium 1 meter long, 1 meter wide, and 1 meter deep is full of water. Find
the work needed to pump half of the water out of the aquarium. (Use 9.8 sm2 for g and the
kg
fact that density of water is 1000 m
3)
Solution: Let y measure depth (in meters) below the top of the tank so that y = 1 at the
bottom of the tank and y = 0 at the top of the tank. A slice of the tank has volume 1 · 1 · dy
and weighs ρgV = ρg · dy. If the slice is at depth y, the work required to move this slice out
of the tank is ρg · y · dy. The work required to empty half of the water from the tank is thus
1
Z 1
Z 1
2
2
1 2 2
1
9.8 · 1000
W =
ρg · y dy = ρg
y dy = ρg y
= ρg =
= 1225 J.
2
8
8
0
0
0