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

Section 15.7 Triple Integrals in Cylindrical Coordinates 1043
z
2
x 2
FIGURE 9
z=2
z=œ„„„„„ ≈+¥
2
y
ExamplE 4 Evaluate y 2 22 ys42x 2
2s42x 2 y2 sx 2 1y 2 sx 2 1 y 2 d dz dy dx.
SOLUTION This iterated integral is a triple integral over the solid region
E − hsx, y, zd | 22 < x < 2, 2s4 2 x 2 < y < s4 2 x 2 , sx 2 1 y 2 < z < 2j
and the projection of E onto the xy-plane is the disk x 2 1 y 2 < 4. The lower sur face of
E is the cone z − sx 2 1 y 2 and its upper surface is the plane z − 2. (See Fig ure 9.)
This region has a much simpler description in cylindrical coordinates:
Therefore we have
E − hsr, , zd | 0 < < 2, 0 < r < 2, r < z < 2j
y 2 22 ys42x2 2s42x 2 y2 sx 2 1y 2
sx 2 1 y 2 d dz dy dx − y y sx 2 1 y 2 d dV
E
− y 2
0
y 2
0 y2 r
r 2 r dz dr d
− y 2
d y 2
r 3 s2 2 rd dr
0
0
− 2 f 1 2 r 4 2 1 5 r 5 g 0
2
−
16
5 ■
1–2 Plot the point whose cylindrical coordinates are given.
Then find the rectangular coordinates of the point.
1. (a) s4, y3, 22d (b) s2, 2y2, 1d
2. (a) ss2 , 3y4, 2d (b) s1, 1, 1d
3–4 Change from rectangular to cylindrical coordinates.
3. (a) s21, 1, 1d (b) s22, 2s3 , 3d
4. (a) (2s2 , s2 , 1)
(b) s2, 2, 2d
5–6 Describe in words the surface whose equation is given.
5. r − 2
6. − y6
7–8 Identify the surface whose equation is given.
7. r 2 1 z 2 − 4 8. r − 2 sin
9–10 Write the equations in cylindrical coordinates.
9. (a) x 2 2 x 1 y 2 1 z 2 − 1 (b) z − x 2 2 y 2
10. (a) 2x 2 1 2y 2 2 z 2 − 4 (b) 2x 2 y 1 z − 1
;
11–12 Sketch the solid described by the given inequalities.
11. r 2 < z < 8 2 r 2
12. 0 < < y2, r < z < 2
13. A cylindrical shell is 20 cm long, with inner radius 6 cm
and outer radius 7 cm. Write inequalities that describe the
shell in an appropriate coordinate system. Explain how you
have positioned the coordinate system with respect to the
shell.
14. Use a graphing device to draw the solid enclosed by the
paraboloids z − x 2 1 y 2 and z − 5 2 x 2 2 y 2 .
15–16 Sketch the solid whose volume is given by the integral
and evaluate the integral.
15. y y2
2y2 y2 0 yr2 0
r dz dr d 16. y 2
17–28 Use cylindrical coordinates.
y r
0 y2 0 0
r dz d dr
17. Evaluate yyy E sx 2 1 y 2 dV, where E is the region that lies
inside the cylinder x 2 1 y 2 − 16 and between the planes
z − 25 and z − 4.
18. Evaluate yyy E
z dV, where E is enclosed by the paraboloid
z − x 2 1 y 2 and the plane z − 4.
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