258
CHAPTER 31
Fig. 31-8
The area
The moment about the Jt-axis is
To compute
we use integration by parts.
The moment about the y-axis is
Hence,
Thus,
Thus,
Let
Then
In
In
In
In
In
In
31.31
The region under y = 4-x
2
in the first quadrant (Fig. 31-9).
The area
The moment about the y-axis is
The moment about the x-axis is
Thus,
Then
Let
H=4 — y,
du — —dy.
Thus,
Fig. 31-9
Fig. 31-10
31.32
The region between y = x
2
and x = y
2
(Fig. 31-10).
The area
The moment about the y-axis is M =
Hence,
By symmetry about the line y = x,
31.33
Use Pappus's theorem to find the volume of a torus obtained by revolving a circle of radius a about a line in its
plane at a distance b from its center (b > a).
Pappus's theorem states that the volume of a solid generated by revolving a region 91 about a line Jifnot passing
through the region is equal to the product of the area A of &t and the distance d traveled around the line by its
centroid. In this case, A = ira
2 ; the centroid is the center of the circle (by symmetry), so that d = 2irb.
Hence, the volume V= ira
2 • 2irb = 2tr
2 a
2 b.
31.34
Use Pappus's theorem to find the volume of a right circular cone of height h and radius of base b.
X
2 )dx = 2x
2 -\x
4 ]
2
0 = 8-4 = 4.