James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 8.3 Applications to Physics and Engineering 565
We end this section by showing a surprising connection between centroids and volumes
of revolution.
This theorem is named after the Greek
mathematician Pappus of Alexandria,
who lived in the fourth century ad.
Theorem of Pappus Let 5 be a plane region that lies entirely on one side of a
line l in the plane. If 5 is rotated about l, then the volume of the resulting solid is
the product of the area A of 5 and the distance d traveled by the centroid of 5.
Proof We give the proof for the special case in which the region lies between
y − f sxd and y − tsxd as in Figure 13 and the line l is the y-axis. Using the method of
cylindrical shells (see Section 6.3), we have
V − y b
2xf f sxd 2 tsxdg dx
a
− 2 y b
xf f sxd 2 tsxdg dx
a
− 2sxAd (by Formulas 9)
− s2 xdA − Ad
where d − 2 x is the distance traveled by the centroid during one rotation about the
y-axis.
n
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 s. rd from the center of the circle. Find the volume of
the torus.
SOLUTION The circle has area A − r 2 . By the symmetry principle, its centroid is its
center and so the distance traveled by the centroid during a rotation is d − 2R. Therefore,
by the Theorem of Pappus, the volume of the torus is
V − Ad − s2Rdsr 2 d − 2 2 r 2 R
n
The method of Example 7 should be compared with the method of Exercise 6.2.63.
1. 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 kgym 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.
3.
8 ft
3 ft
2 ft
4.
5 ft
2 ft
10 ft
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