INTEGRALS FOR SURFACE AREA, WORK, CENTROIDS
255
Fig. 31-1
31.15
Find the surface area of a cap of a sphere with radius a determined by a plane at a distance b from the center.
The cap is generated by revolving about the jc-axis the region in the first quadrant under
between x = b and x = a. Since x
2 + y
2 = a
2 , 2x + 2yy' = Q,
Hence,
WORK
31.16
A spring with a natural length of 10 inches is stretched | inch by a 12-pound force. Find the work done in
stretching the spring from 10 to 18 inches.
We use Hooke's law: The spring pulls back with a restoring force of F = kx pounds, where the spring is
stretched x inches beyond its natural length, and k is a constant. Then, 12=j/t, A: = 24. F=24x, and the
work W= J 0
8 F dx = J 0
8 24x dx = Ux
2 ]
8
0 = 12(64) = 768 in • Ib = 64 ft • Ib.
31.17
A spring supporting a railroad car has a natural length of 12 inches, and a force of 8000 pounds compresses it
2 inch. Find the work done in compressing it from 12 to 9 inches.
Hooke's law also holds for compression. Then F=kx, 8000= \k, k = 16,000. So the work W =
J 0
3
16,000 x dx = 8000*
2 ]
3
0 = 8000(9) = 72,000 in • Ib = 6000 ft • Ib.
31.18
A bucket, weighing 5 pounds when empty, is loaded with 60 pounds of sand, and then lifted (at constant speed) 10
feet. Sand leaks out of a hole in the bucket at a uniform rate, and a third of the sand is lost by the end of the
lifting. Find the work done in the lifting process.
_ Let x be the height above the initial position. The sand leaks out at the uniform rate of 2 pounds per foot.
The force being exerted when the bucket is at position x is 65 — 2x, the weight of the load. Hence, the work
W = J 0
10 (65 - 2x) dx = (65x - x
2 ) ]
1
0 ° = 650 - 100 = 550 ft • Ib.
31.19
A 5-lb monkey is attached to the end of a 30-ft hanging rope that weighs 0.2 Ib/ft. The monkey climbs the rope
to the top. How much work has it done?
At height x above its initial position, the monkey must exert a force 5 + 0.2* to balance its own weight and
the weight of rope below that point. Hence, the work W= Jo° (5 + 0.2x) dx = 5* + O.lx
2 ]l° = 150 + 90 =
240ft-lb.
31.20
A conical tank, 10 meters deep and 8 meters across at the top, is filled with water to a depth of 5 meters. The
tank is emptied by pumping the water over the top edge. How much work is done in the process?
Fig. 31-2
255
Fig. 31-1
31.15
Find the surface area of a cap of a sphere with radius a determined by a plane at a distance b from the center.
The cap is generated by revolving about the jc-axis the region in the first quadrant under
between x = b and x = a. Since x
2 + y
2 = a
2 , 2x + 2yy' = Q,
Hence,
WORK
31.16
A spring with a natural length of 10 inches is stretched | inch by a 12-pound force. Find the work done in
stretching the spring from 10 to 18 inches.
We use Hooke's law: The spring pulls back with a restoring force of F = kx pounds, where the spring is
stretched x inches beyond its natural length, and k is a constant. Then, 12=j/t, A: = 24. F=24x, and the
work W= J 0
8 F dx = J 0
8 24x dx = Ux
2 ]
8
0 = 12(64) = 768 in • Ib = 64 ft • Ib.
31.17
A spring supporting a railroad car has a natural length of 12 inches, and a force of 8000 pounds compresses it
2 inch. Find the work done in compressing it from 12 to 9 inches.
Hooke's law also holds for compression. Then F=kx, 8000= \k, k = 16,000. So the work W =
J 0
3
16,000 x dx = 8000*
2 ]
3
0 = 8000(9) = 72,000 in • Ib = 6000 ft • Ib.
31.18
A bucket, weighing 5 pounds when empty, is loaded with 60 pounds of sand, and then lifted (at constant speed) 10
feet. Sand leaks out of a hole in the bucket at a uniform rate, and a third of the sand is lost by the end of the
lifting. Find the work done in the lifting process.
_ Let x be the height above the initial position. The sand leaks out at the uniform rate of 2 pounds per foot.
The force being exerted when the bucket is at position x is 65 — 2x, the weight of the load. Hence, the work
W = J 0
10 (65 - 2x) dx = (65x - x
2 ) ]
1
0 ° = 650 - 100 = 550 ft • Ib.
31.19
A 5-lb monkey is attached to the end of a 30-ft hanging rope that weighs 0.2 Ib/ft. The monkey climbs the rope
to the top. How much work has it done?
At height x above its initial position, the monkey must exert a force 5 + 0.2* to balance its own weight and
the weight of rope below that point. Hence, the work W= Jo° (5 + 0.2x) dx = 5* + O.lx
2 ]l° = 150 + 90 =
240ft-lb.
31.20
A conical tank, 10 meters deep and 8 meters across at the top, is filled with water to a depth of 5 meters. The
tank is emptied by pumping the water over the top edge. How much work is done in the process?
Fig. 31-2