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Stewart_Calc_7ET ch08sec03
MULTIPLE CHOICE
1. A gate in an irrigation canal is constructed in the form of a trapezoid ft wide at the bottom, ft
wide at the top, and 2 ft high. It is placed vertically in the canal, with the water extending to its top.
Find the hydrostatic force on one side of the gate..
a.
lb
b.
lb
c.
lb
d.
lb
e.
None of these
2. You are given the shape of the vertical ends of a trough that is completely filled with water. Find the
force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)
a.
104 lb
b.
1040 lb
c.
208 lb
d.
520 lb
3. You are given the shape of the vertical ends of a trough that is completely filled with water. Find the
force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)
a.
2662.4 lb
b.
1331.2 lb
c.
665.6 lb
d.
332.8 lb
5 ft
4ft
4ft
4. A trough has vertical ends that are equilateral triangles with sides of length 2 ft. If the trough is filled
with water to a depth of 1 ft, find the force exerted by the water on one end of the trough. Round to
one decimal place. (The weight density of water is 62.4 lb/ft3.)
a.
31.2 lb
b.
12.0 lb
c.
62.4 lb
d.
6.0 lb
5. Find the centroid of the region bounded by the graphs of the given equations.
a.
b.
c.
d.
6. The masses are located at the point . Find the moments and and the center of mass of
the system.
;
a.
b.
c.
d.
e.
7. A large tank is designed with ends in the shape of the region between the curves and ,
measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of ft with
gasoline. (Assume that the density of the gasoline is 42.0 lb/ .)
a.
lb
b.
6,918 lb
c.
3,855 lb
d.
5,785 lb
e.
4,850 lb
f.
2,683 lb
8. A trough is filled with a liquid of density 855 kg/ . The ends of the trough are equilateral triangles
with sides m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.
a.
b.
c.
d.
e.
9. Find the centroid of the region bounded by the given curves.
a.
b.
c.
d.
e.
None of these
10. Find the centroid of the region bounded by the given curves.
a.
b.
c.
d.
e.
11. A vertical plate is submerged in water (the surface of the water coincides with the x-axis). Find the
force exerted by the water on the plate. (The weight density of water is 62.4 lb/ft3.)
(ft)
a.
62.4 lb
b.
873.6 lb
c.
436.8 lb
d.
124.8 lb
12. A rectangular tank has width 4 ft, height 4 ft, and length 7 ft. It is filled with equal volumes of water
and oil. The oil has a weight density of 50 lb/ft3 and floats on the water. Find the force exerted by
the mixture on one end of the tank. (The weight density of water is 62.4 lb/ft3.)
a.
1897.6 lb
b.
1699.2 lb
(ft)
12345x
1
–1
–2
–3
–4
y
c.
899.2 lb
d.
1798.4 lb
13. Find the centroid of the region bounded by the graphs of the given equations.
a.
b.
c.
d.
14. Use the Theorem of Pappus to find the volume of the solid obtained by revolving the region bounded
by the graphs of and about the y-axis.
a.
b.
c.
d.
NUMERIC RESPONSE
1. Find the coordinates of the centroid for the region bounded by the curves , x = 0,
and y = .
2. A swimming pool is 10 ft wide and 36 ft long and its bottom is an inclined plane, the shallow end
having a depth of ft and the deep end, 12 ft. If the pool is full of water, find the hydrostatic force on
the shallow end. (Use the fact that water weighs 62.5 lb/ .)
648
144
864
108
3. Find the exact coordinates of the centroid.
4. Calculate the center of mass of the lamina with density = .
5. Find the centroid of the region bounded by the given curves.
6. Find the centroid of the region bounded by the curves.
7. Find the center of mass of a lamina in the shape of a quarter-circle with radius with density = .
8. Find the centroid of the region shown, not by integration, but by locating the centroids of the
rectangles and triangles and using additivity of moments.
SHORT ANSWER
1. An aquarium is 4 ft long, 3 ft wide, and 2 ft deep. If the aquarium is filled with water, find the force
exerted by the water (a) on the bottom of the aquarium, (b) on the longer side of the aquarium, and (c)
on the shorter side of the aquarium. (The weight density of water is 62.4 lb/ft3.)
2. You are given the shape of the vertical ends of a trough that is completely filled with water. Find the
force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)
3. A cylindrical drum of diameter 2 ft and length 6 ft is lying on its side, submerged in water 16 ft deep.
Find the force exerted by the water on one end of the drum to the nearest pound. (The weight density
of water is 62.4 lb/ft3.)
4. Find the centroid of the region bounded by the graphs of and .
6ft
3 ft
2 ft
5. You are given the shape of the vertical ends of a trough that is completely filled with water. Find the
force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)
6. A vertical plate is submerged in water (the surface of the water coincides with the x-axis). Find the
force exerted by the water on the plate. (The weight density of water is 62.4 lb/ft3.)
(ft)
7. Find the centroid of the region bounded by the graphs of the given equations.
5 ft
7ft
10 ft
(ft)
12345x
1
–1
–2
–3
–4
y
8. You are given the shape of the vertical ends of a trough that is completely filled with water. Find the
force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)
9. A vertical plate is submerged in water (the surface of the water coincides with the x-axis). Find the
force exerted by the water on the plate. (The weight density of water is 62.4 lb/ft3.)
(ft)
10. Find the center of mass of the system comprising masses mk located at the points Pk in a coordinate
plane. Assume that mass is measured in grams and distance is measured in centimeters.
m1 = 3, m2 = 4, m3 = 5
P1 (–3, 5), P2 (3, 4), P3 (–4, 1)
11. Find the centroid of the region shown in the figure.
8ft
3 ft
2 ft
(ft)
12345x
1
–1
–2
–3
–4
y
12. Find the center of mass of the lamina of the region shown if the density of the circular lamina is five
times that of the rectangular lamina.
13. Find the centroid of the region bounded by the graphs of and .
–1
1
1–1 x
y
2–2 x
1
–1
y
