taller calculo b

1. we would have obtained the integral
EXAMPLE 9 A wedge is cut out of a circular cylinder of radius 4 by two planes. One
plane is perpendicular to the axis of the cylinder. The other intersects the first at an angle
of 30 along a diameter of the cylinder. Find the volume of the wedge.
SOLUTION If we place the -axis along the diameter where the planes meet, then the
base of the solid is a semicircle with equation , . A cross-
section perpendicular to the -axis at a distance from the origin is a triangle ,
as shown in Figure 17, whose base is and whose height is
. Thus the cross-sectional area is
and the volume is
For another method see Exercise 64. M
෇
128
3s3
෇
1
s3 y
4
0
͑16 Ϫ x2
͒ dx ෇
1
s3 ͫ16x Ϫ
x3
3 ͬ0
4
V ෇ y
4
Ϫ4
A͑x͒ dx ෇ y
4
Ϫ4
16 Ϫ x2
2s3
dx
෇
16 Ϫ x2
2s3
A͑x͒ ෇ 1
2 s16 Ϫ x2 ؒ
1
s3
s16 Ϫ x2
ԽBCԽ ෇ y tan 30Њ ෇ s16 Ϫ x2͞s3
y ෇ s16 Ϫ x2
ABCxx
Ϫ4 ഛ x ഛ 4y ෇ s16 Ϫ x2
x
Њ
V ෇ y
h
0
L2
h2 ͑h Ϫ y͒2
dy ෇
L2
h
3
430 |||| CHAPTER 6 APPLICATIONS OF INTEGRATION
y=œ„„„„„„16-≈
x
y0
A
B
C
4
FIGURE 17
A B
C
y
30°
10. , , ; about the -axis
, ; about
12. , , ; about
13. , ; about
14. , , , ; about
15. , ; about
16. , ; about
17. , ; about
18. , , , ; about x ෇ 1x ෇ 4x ෇ 2y ෇ 0y ෇ x
x ෇ Ϫ1x ෇ y2
y ෇ x2
x ෇ 2y ෇ sxy ෇ x
x ෇ 1x ෇ 1x ෇ y2
y ෇ Ϫ1x ෇ 3x ෇ 1y ෇ 0y ෇ 1͞x
y ෇ 1y ෇ 3y ෇ 1 ϩ sec x
y ෇ 2x ෇ 2y ෇ 1y ෇ eϪx
y ෇ 1y ෇ sxy ෇ x11.
yy ෇ 0x ෇ 2y ෇
1
4 x21–18 Find the volume of the solid obtained by rotating the region
bounded by the given curves about the specified line. Sketch the
region, the solid, and a typical disk or washer.
1. , , , ; about the -axis
2. , ; about the -axis
3. , , , ; about the -axis
4. , , , ; about the -axis
5. , , ; about the -axis
6. , , , ; about the -axis
, , ; about the -axis
8. , ; about the -axis
, ; about the -axisyx ෇ 2yy2
෇ x9.
xy ෇ 5 Ϫ x2
y ෇
1
4 x2
xx ജ 0y ෇ xy ෇ x37.
yx ෇ 0y ෇ 2y ෇ 1y ෇ ln x
yy ෇ 9x ෇ 0x ෇ 2sy
xx ෇ 4x ෇ 2y ෇ 0y ෇ s25 Ϫ x2
xy ෇ 0x ෇ 2x ෇ 1y ෇ 1͞x
xy ෇ 0y ෇ 1 Ϫ x2
xx ෇ 2x ෇ 1y ෇ 0y ෇ 2 Ϫ
1
2 x
EXERCISES6.2

2. 44.
45. A CAT scan produces equally spaced cross-sectional views of
a human organ that provide information about the organ other-
wise obtained only by surgery. Suppose that a CAT scan of a
human liver shows cross-sections spaced 1.5 cm apart. The
liver is 15 cm long and the cross-sectional areas, in square
centimeters, are 0, 18, 58, 79, 94, 106, 117, 128, 63, 39, and
0. Use the Midpoint Rule to estimate the volume of the liver.
46. A log 10 m long is cut at 1-meter intervals and its cross-
sectional areas (at a distance from the end of the log) are
listed in the table. Use the Midpoint Rule with to esti-
mate the volume of the log.
47. (a) If the region shown in the figure is rotated about the
-axis to form a solid, use the Midpoint Rule with
to estimate the volume of the solid.
(b) Estimate the volume if the region is rotated about the
-axis. Again use the Midpoint Rule with .
48. (a) A model for the shape of a bird’s egg is obtained by
rotating about the -axis the region under the graph of
Use a CAS to find the volume of such an egg.
(b) For a Red-throated Loon, , , ,
and . Graph and find the volume of an egg of
this species.
49–61 Find the volume of the described solid .
A right circular cone with height and base radius
50. A frustum of a right circular cone with height , lower base
radius , and top radius
R
h
r
rR
h
rh49.
S
fd ෇ 0.54
c ෇ 0.1b ෇ 0.04a ෇ Ϫ0.06
f͑x͒ ෇ ͑ax3
ϩ bx2
ϩ cx ϩ d͒s1 Ϫ x2
x
CAS
n ෇ 4y
0 4
4
102 86
2
y
x
n ෇ 4x
n ෇ 5
xA
␲ y
␲͞2
0
͓͑1 ϩ cos x͒2
Ϫ 12
͔ dx␲ y
1
0
͑y4
Ϫ y8
͒ dy43.19–30 Refer to the figure and find the volume generated by
rotating the given region about the specified line.
19. about 20. about
21. about 22. about
23. about 24. about
25. about 26. about
27. about 28. about
29. about 30. about
31–36 Set up, but do not evaluate, an integral for the volume of
the solid obtained by rotating the region bounded by the given
curves about the specified line.
31.
32. , ; about
33. , , ; about
34. , , ; about
35. , ; about
36. , , ; about
; 37–38 Use a graph to find approximate -coordinates of the
points of intersection of the given curves. Then use your calcula-
tor to find (approximately) the volume of the solid obtained by
rotating about the -axis the region bounded by these curves.
37. ,
38.
39–40 Use a computer algebra system to find the exact volume
of the solid obtained by rotating the region bounded by the given
curves about the specified line.
39. , , ;
40. , ;
41–44 Each integral represents the volume of a solid. Describe
the solid.
41. 42. ␲ y
5
2
y dy␲ y
␲͞2
0
cos2
x dx
about y ෇ 3y ෇ xe1Ϫx͞2
y ෇ x
about y ෇ Ϫ10 ഛ x ഛ ␲y ෇ 0y ෇ sin2
x
CAS
y ෇ 3 sin͑x2
͒, y ෇ ex͞2
ϩ eϪ2x
y ෇ x4
ϩ x ϩ 1y ෇ 2 ϩ x2
cos x
x
x
y ෇ 40 ഛ x ഛ 2␲y ෇ 2 Ϫ cos xy ෇ cos x
x ෇ Ϫ2x ෇ 3x2
Ϫ y2
෇ 1
y ෇ Ϫ20 ഛ x ഛ ␲y ෇ sin xy ෇ 0
y ෇ 10 ഛ x ഛ ␲y ෇ sin xy ෇ 0
x ෇ 108x Ϫ y ෇ 16y ෇ ͑x Ϫ 2͒4
y ෇ tan3
x, y ෇ 1, x ෇ 0; about y ෇ 1
BC᏾3AB᏾3
OC᏾3OA᏾3
BC᏾2AB᏾2
OC᏾2OA᏾2
BC᏾1AB᏾1
OC᏾1OA᏾1
O x
y
T™
y=˛
T£
T¡
B(1,€1)
A(1,€0)
y=œ„x
C(0,€1)
SECTION 6.2 VOLUMES |||| 431
x (m) A ( ) x (m) A ( )
0 0.68 6 0.53
1 0.65 7 0.55
2 0.64 8 0.52
3 0.61 9 0.50
4 0.58 10 0.48
5 0.59
m2
m2