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

568 Chapter 8 Further Applications of Integration
46. The solid obtained by rotating the triangle with vertices
s2, 3d, s2, 5d, and s5, 4d about the x-axis
47. The centroid of a curve can be found by a process similar to
the one we used for finding the centroid of a region. If C
is a curve with length L, then the centroid is sx, yd where
x − s1yLd y x ds and y − s1yLd y y ds. Here we assign appropriate
limits of integration, and ds is as defined in Sections 8.1
and 8.2. (The centroid often doesn’t lie on the curve itself. If
the curve were made of wire and placed on a weightless board,
the centroid would be the balance point on the board.) Find the
centroid of the quarter-circle y − s16 2 x 2 , 0 < x < 4.
48. The Second Theorem of Pappus is in the same spirit as
Pappus’s Theorem on page 565, but for surface area rather
than volume: Let C be a curve that lies entirely on one side of
a line l in the plane. If C is rotated about l, then the area of the
resulting surface is the product of the arc length of C and the
distance traveled by the centroid of C (see Exercise 47).
(a) Prove the Second Theorem of Pappus for the case where C
is given by y − f sxd, f sxd > 0, and C is rotated about the
x-axis.
(b) Use the Second Theorem of Pappus to compute the surface
area of the half-sphere obtained by rotating the curve from
Exercise 47 about the x-axis. Does your answer agree with
the one given by geometric formulas?
49. Use the Second Theorem of Pappus described in Exercise 48 to
find the surface area of the torus in Example 7.
50. Let 5 be the region that lies between the curves
y − x m y − x n 0 < x < 1
where m and n are integers with 0 < n , m.
(a) Sketch the region 5.
(b) Find the coordinates of the centroid of 5.
(c) Try to find values of m and n such that the centroid lies
outside 5.
51. Prove Formulas 9.
discovery Project
complementary coffee cups
Suppose you have a choice of two coffee cups of the type shown, one that bends outward and one
inward, and you notice that they have the same height and their shapes fit together snugly. You
wonder which cup holds more coffee. Of course you could fill one cup with water and pour it
into the other one but, being a calculus student, you decide on a more mathematical approach.
Ignoring the handles, you observe that both cups are surfaces of revolution, so you can think of
the coffee as a volume of revolution.
y
h
x=k
A¡
A
x=f(y)
Cup A
Cup B
0
k
x
1. Suppose the cups have height h, cup A is formed by rotating the curve x − f syd about the
y-axis, and cup B is formed by rotating the same curve about the line x − k. Find the value
of k such that the two cups hold the same amount of coffee.
2. What does your result from Problem 1 say about the areas A 1 and A 2 shown in the figure?
3. Use Pappus’s Theorem to explain your result in Problems 1 and 2.
4. Based on your own measurements and observations, suggest a value for h and an equation for
x − f syd and calculate the amount of coffee that each cup holds.
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