[ill x = j .JY (y - 3),

§ EXERCISES
I. An aquarium 5 ft long, 2 ft wide, and 3 ft deep is full of
water. Find (a) the hydrostatic pressure on the bottom of the
aquarium, (b) the hydrostatic force on the bottom, and (c) the
hydrostatic force on one end of the aquarium.
2. A tank is 8 m long, 4 m wide, 2 m high, and contains kerosene
with density 820 kg/m 3 to a depth of 1.5 m. Find (a) the hydrostatic
pressure on the bottom of the tank, (b) the hydrostatic
force on the bottom, and (c) the hydrostatic force on one end
of the tank.
3-11 A vertical plate is submerged (or partially submerged) in
water and has the indicated shape. Explain how to approximate the
hydrostatic force against one side of the plate by a Riemann sum.
Then express the force as an integral and evaluate it.
v = f: 217X[j(X) - g(X)] dx
= 217 s: x [j(x) - g(x)] dx
where d = 217X is the distance traveled by the centroid during one rotation about the
y-axis. []
L!J EXAMPLE 7 A torus is formed by rotating a circle of radius r about a line in the plane
of the circle that is a distance R (> r) from the center of the circle. Find the volume of
the torus.
SOLUTION The circle has area A = l7r 2 . By the symmetry principle, its centroid is its center
and so the distance traveled by the centroid during a rotation is d = 217R. Therefore,
by the Theorem of Pappus, the volume of the torus is
The method of Example 7 should be compared with the method of Exercise 63 in
Section 6.2.