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

1014 Chapter 15 Multiple Integrals
y
(x-1)@+¥=1
(or r=2 cos ¨)
ExamplE 4 Find the volume of the solid that lies under the paraboloid z − x 2 1 y 2 ,
above the xy-plane, and inside the cylinder x 2 1 y 2 − 2x.
SOLUTION The solid lies above the disk D whose boundary circle has equation
x 2 1 y 2 − 2x or, after completing the square,
0
D
1 2
x
sx 2 1d 2 1 y 2 − 1
(See Figures 9 and 10.)
In polar coordinates we have x 2 1 y 2 − r 2 and x − r cos , so the boundary circle
becomes r 2 − 2r cos , or r − 2 cos . Thus the disk D is given by
FIGURE 9
D − hsr, d | 2y2 < < y2, 0 < r < 2 cos j
z
and, by Formula 3, we have
V − y sx 2 1 y 2 d dA − y y2 cos
y2 r 2 r dr d −
2y2 0
D
y
2y2F r 4 2 cos
d
4G0
− 4 y y2
2y2 cos4 d − 8 y y2
cos 4 y2
d − 8 y
0
0
S
1 1 cos 2D
2
2
d
x
y
− 2 y y2
f1 1 2 cos 2 1 1 2 s1 1 cos 4dg d
0
FIGURE 10
− 2f 3 2 1 sin 2 1 1 8 sin 4g 0
y2
− 2 S 3 2DS 2D − 3 2
■
1–4 A region R is shown. Decide whether to use polar coordinates
or rectangular coordinates and write yy R
f sx, yd dA as an iterated
integral, where f is an arbitrary continuous function on R.
1.
R
y
0
5
2
2
5
x
2.
y
1
R
0 _1 1
x
5 –6 Sketch the region whose area is given by the integral and
evaluate the integral.
5. y 3y4
y 2
y4 1
r dr d
6. y y2
sin
y2 r dr d
0
7–14 Evaluate the given integral by changing to polar coordinates.
7. yy D
x 2 y dA, where D is the top half of the disk with center the
origin and radius 5
3.
y
4.
y
8. yy R
s2x 2 yd dA, where R is the region in the first quadrant
enclosed by the circle x 2 1 y 2 − 4 and the lines x − 0
and y − x
9. yy R
sinsx 2 1 y 2 d dA, where R is the region in the first quadrant
between the circles with center the origin and radii 1 and 3
_1
R
0
_1
1
x
R
0 3
_3
x
10. y y
R
y 2
x 2 2
dA, where R is the region that lies between the
1 y
circles x 2 1 y 2 − a 2 and x 2 1 y 2 − b 2 with 0 , a , b
11. yy D
e 2x2 2y 2 dA, where D is the region bounded by the semi-
circle x − s4 2 y 2 and the y-axis
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.