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Unformatted text preview: Section 7.3
2. In each case, the width of the cross section is w
w
, so A(x)
2 r 2, where r (a) A
(b) A s 2, where s (c) A s 2, where s w, so A(x)
w 2 x. w2 x. 2 w 2 8. A cross section has width w 4x.
/3 w2 , so A(x) 2 V 2x. /3 4 2 /3 32
w (see Quick Review Exercise 5), so (d) A 3
4 x)2 (2 4 4 x2 2x dx
0 4 x and area 2 2 2 w s2 (sec2 x tan x 2 sec x tan x r x 2) (2 1 x ) dx (x 2x 2 2 2 2x x ) . The volume is 2 23
x
3 w2 21 1 4 x 2) dx (1 1 1 2 x 2 and area 4x 2 x ) dx 1 2 2 (1 1 1
1 16
.
3 x 2 and area x ) dx 2x 1 7. A cross section has width w
32
w (a) A(x) 13
x
3 13
x
3 1
1 8
.
3 2 sin x. 54
y dy
4 y5 4 8.
0 10. A cross section has width w
12
s
21 12
w
2 1 y 2 and area y 2). The volume is 2(1 y 2) dy 2 2y 13
y
3 1
1 8
.
3 11. (a) The volume is the same as if the square had moved
without twisting: V Ah s 2h.
(b) Still s 2h: the lateral distribution of the square cross
sections doesn’t affect the volume. That’s Cavalieri’s
Volume Theorem. from y 3 sin x dx 12
2 6 to y 12, for a diameter of 6 and a radius of 3, the solid has the same cross sections as the right 0 3 54
y . The volume is
42 2 12. Since the diameter of the circular base of the solid extends 3 sin x, and 4 V w2 2(1 x 2). The volume is 2(1 2 5y 2 and area 9. A cross section has width w 1 2 w 21 2
.
3 x 1 6. A cross section has width w tan x)2 dx, which by same method as (sec x x 2). The volume is 4(1 x 2) dx 2(1 6 tan x)2, and (sec x in part (a) equals 4 3 1 0 1 2 /3 r2 5. A cross section has width w w2 /3 V 1) dx 16
.
15 s2 3 6 . 6 s2 22 (1 4 15
x
5 x2 (b) A(x) w2
2 2 1 A(x) /3 2 1 4(1 /3 1
x
2 sec x 3 2 16. 22 s2 x
/3 2 and area A(x) 1 tan x 2x. The volume is 4. A cross section has width w A(x) tan2 x) dx 2 sec x tan x /3 3 (1 tan x)2, and (sec x tan x)2 dx (sec x 0 1 4 tan x. /3 3x. 3. A cross section has width w
A(x) 4 4 A(x) w2
2 r2 (a) A(x) sec x 301 sin x dx circular cone. The volumes are equal by Cavalieri’s 0 3 Theorem. cos x
0 2 s2 (b) A(x)
V 13. The solid is a right circular cone of radius 1 and height 2. 3.
w2 4 sin x dx
0 V 4 sin x, and
4 sin x dx
0 4 cos x 8. 1
Bh
3 1
( r 2)h
3 1
( 12)2
3 2
3 14. The solid is a right circular cone of radius 3 and height 2. 0 V 1
Bh
3 1
( r 2)h
3 1
( 32)2
3 6 302 Section 7.3 15. A cross section has radius r
r2 A( y )
1 tan2 4 0 tan2
y dy 4
4 tan 4 y and area 19. y . The volume is
1 tan 4 4 y y
[ 6, 6] by [ 4, 4] 0 1
The solid is a sphere of radius r 4 . 43
r
3 16. A cross section has radius r sin x cos x and area A(x) r2 from x 3. The volume is 36 . 20. sin2 x cos2 x. The shaded region extends 0 to where sin x cos x drops back to 0, i.e., where
[ 0.5, 1.5] by [ 0.5, 0.5] x 2 2 cos2 x . Now, since cos 2x 1, we know The parabola crosses the line y 1 cos 2x
cos x
and since cos 2x 1 2 sin2 x, we
2
/2
1 cos 2x
know sin2 x
.
sin2 x cos2 x dx
0
2
/2
1 cos 2x 1 cos 2x
dx
0
2
2
2 /2 4 (1 0 /2 4
8 1 0 x /2 cos2 2x) dx
cos 4x
dx
2 1
sin 4x
4 4 0 0 x(1 x) r2 (x 2 2x 3 x 4). 14
x
2 15
x
5 The volume is
1 cos 4x) dx 1. A cross x and area (x 2 2x 3 13
x
3 x 4) dx 0 (1 0 or x 2 x 2)2 (x 0 when 0, i.e., when x
x A(x) /2 8 x section has radius r sin2 2x dx /2
0 x 2 1
0 30 . 21. 2 8 2 0 0 16 . 17. [ 1, 2] by [ 1, 2] Use cylindrical shells: A shell has radius y and height y.
The volume is
1 2 ( y)( y) dy [ 2, 4] by [ 1, 5] 2 0 A cross section has radius r
A(x)
2 r2 0 1 2
.
3 0 22. x4. The volume is
15
x
5 x 4 dx x 2 and area 13
y
3 2
0 32
.
5
[ 1, 3] by [ 1, 3] 18. Use washer cross sections: A washer has inner radius r
outer radius R
1 3 x 2 dx The volume is [ 4, 6] by [ 1, 9] 0 A cross section has radius r
A(x)
2
0 x 6 dx r2 x 3 and area x 6. The volume is
17
x
7 2
0 128
.
7 (R 2 2x, and area A(x)
3 13
x
3 r 2) 1 .
0 3 x 2. x, Section 7.3
23. 303 26. [ 2, 3] by [ 1, 6] [ 1, 5] by [ 3, 1] The curves intersect when x 2 1 x 3, which is when
x 2 x 2 (x 2)(x 1) 0, i.e., when
x
1 or x 2. Use washer cross sections: a washer has
inner radius r x 2 1, outer radius R x 3, and area
A(x)
(R 2 r 2)
[(x 3)2 (x2 1)2]
( x 4 x 2 6x 8). The volume is
2 ( x4 x2 6x The curves intersect where
x
2, which is where
x 4. Use washer cross sections: a washer has inner radius
r
x, outer radius R 2, and area
A(x)
(R 2 r 2)
(4 x).
4 The volume is (4 x) dx 4x 0 12
x
2 4 8
0 27. 8) dx 1 15
x
5 2 13
x
3 32
5 8
3 3x 2 8x
1 12 1
5 16 1
3 3 8 [ 0.5, 1.5] by [ 0.5, 2] 117
.
5 The curves intersect at x 24. radius r 2
r2 A(x) The curves intersect when 4 x
2 x, which is when
x2 x 2 (x 2)(x 1) = 0, i.e., when x
1 or
x 2. Use washer cross sections: a washer has inner radius
r 2 x, outer radius R 4 x 2, and area
A(x)
(R 2 r 2)
[(4 x 2)2 (2 x)2]
(12 4x 9x 2 x 4). 4x [ 1, 3] by [ 1, 3] The curve and horizontal line intersect at x
9x 2 section has radius 2 x 4) dx 1 r2 A(x)
12x
24 2x 2
8 2.301. 28. The volume is
(12 sec x tan x)2. Use NINT to find 0 2 2 2 sec x tan x)2 dx (2 [ 2, 3] by [ 1, 5] sec x tan x and area
( 0.7854 0.7854. A cross section has 15
x
5 3x 3
24 sin x)2 4 (1 4 (1 2 sin x The volume is 32
5 /2 12 2 3 1
5 2 sin x sin2 x) dx 2 cos x 4 (1 108
.
5 1
sin 2x
4 0 4 25. 4 3
x
2
3
4 2 (3 /2
0 8) 29.
3 , 3 by [ 0.5, 2] Use washer cross sections: a washer has inner radius
r sec x, outer radius R
2, and area
A(x)
(R 2 r 2)
(2 sec2 x).
The volume is
/4 (2 sec2 x) dx [ 1, 3] by [ 1.5, 1.5] A cross section has radius r
/4 2x A(y) tan x /4 2
2 2. 2 The volume is
1 5y 2 and area 5 y 4.
1 /4 1 r2 5 y 4 dy 1 . A cross 2 sin x and area 2
1 2 y5 1 2.
1 sin2 x). 304 Section 7.3 30. 35. [ 1, 4] by [ 1, 3] [ 1, 5] by [ 1, 3] A cross section has radius r
r2 A(y)
2 The curved and horizontal line intersect at (4, 2). y 3/2 and area (a) Use washer cross sections: a washer has inner radius y 3. The volume is
14
y
4 y 3 dy 0 2 r 4.
0 x, outer radius R A(x) 31. (R 2 2, and area 2 r) (4 4 (4 0 x) dx x). The volume is
12
x
2 4x 4 8
0 y 2 and area (b) A cross section has radius r
[ 1.2, 3.5] by [ 1, 2.1] r2 A(y) y 4.
2 Use washer cross sections. A washer has inner radius r
outer radius R y (R 2 A(y) 1 ( y2 0 2 32
.
5 0 1, and area r 2) volume is 15
y
5 y 4 dy The volume is 1, 1)2 [(y 13
y
3 2y) dy 0 (c) A cross section has radius r (y2 1]
y2 1
0 2 x and area 2y). The
r2 A(x) 4
.
3 x)2 (2 (4 4 x x). The volume is 32. 4 (4 4 x x) dx 8 3/2
x
3 4x 0 12
x
2 4
0 8
.
3 (d) Use washer cross sections: a washer has inner radius
[ 1.7, 3] by [ 1, 2.1] r
Use cylindrical shells: a shell has radius x and height x. The
1 volume is 2 (x)(x) dx 2 0 13
x
3 1
0 y 2, outer radius R 4 (R 2 A( y ) 2
.
3 (8y 2 r 2) 4, and area [16 83
y
3 15
y
5 y 2)2] (4 y4). 33.
The volume is
2
0 (8y 2 y 4) dy 2 224
15 0 36. [ 2, 4] by [ 1, 5] Use cylindrical shells: A shell has radius x and height x 2.
2 2 (x)(x 2) dx The volume is 2 0 14
x
4 2 8.
0 [ 1, 3] by [ 1, 3] 34. The slanted and vertical lines intersect at (1, 2)
(a) The solid is a right circular cone of radius 1 and
height 2. The volume is
[ 0.5, 1.5] by [ 0.5, 1.5] 1
Bh
3 The curves intersect at x 0 and x shells: a shell has radius x and height
1 is 2 (x)(
0 x x) dx 2 2 5/2
x
5 1. Use cylindrical
x
13
x
3 x. The volume
1
0 2
.
15 1
( r 2)h
3 1
( 12)2
3 2
.
3 (b) Use cylindrical shells: a shell has radius 2
height 2x. The volume is
1 1 2 (2 x)(2x) dx 4 0 (2x x 2) dx 0 4 x2 13
x
3 1
0 8
.
3 x and 305 Section 7.3
40. 37. [ 2, 2] by [ 1, 2] [ 2, 2] by [ 1, 3] The curves intersect at ( 1, 1). x2 (a) A cross section has radius r
r2 A(x) 1 x and area (1 x 2)2 (1 2x 2 2 x 4). 2x 2 (1 1. A shell has radius x and height x 2. The volume is x 2 (x)(2 x 2) dx x 13
x
3 x2 2 0 x 4) dx 23
x
3 x 1 1 5
.
6 0 16
.
15 1 y and
[ 1, 5] by [ 1, 3] height 2 y. The volume is
1 1 y)(2 y) dy 14
x
4 41. 1 15
x
5 (b) Use cylindrical shells: a shell has radius 2 2 (2 x at x 1 The volume is
1 2 2 4 (2 0 y y 3/2) dy 0 4 3/2
y
3 4 2 5/2
y
5 1 A shell has radius x and height
4 56
.
15 0 2 (x)( x) dx 2 5/2
x
5 2 0 x. The volume is 4 128
.
5 0 42.
(c) Use cylindrical shells: a shell has radius y 1 and height 2 y. The volume is
1 1 2 (y 1)(2 y) dy ( y3/2 4 0 2 5/2
y
5 4
38. (a) A cross section has radius r
r2 A(x)
b y) dy 0 2 h1
0 h2 1
x2
dx
b 2 3/2
y
3 1
0 [ 2, 2] by [ 2, 2] 64
.
15 x
and area
b h1 x2
. The volume is
b
b
b
x3
h2
1
bh 2.
3
3
b0 The functions intersect where 2x x
. The volume is
b
b
x
2 (x)h 1
dx 2 h
b
0 x (2x 1) x 2x x2
dx
b
b
x3
b 2h.
3
3b 0 x
0 1
2 2 h x2 1. The volume is 1 1 2 (x)( x 2x 1) dx (x 3/2 2 0 2x 2 x) dx 23
x
3 12
x
2 0 2 5/2
x
5 2
7
.
15 43. A shell has height 12( y 2
b x, i.e., at x A shell has radius x and height (b) Use cylindrical shells: a shell has radius x and height
h1 1 (a) A shell has radius y. The volume is
1 2 ( y)12( y 2 y 3) dy 1 ( y3 24 0 y 4) dy 0 24
(b) A shell has radius 1
1 y)12( y 2 2 (1
0 1 ( y4 24 14
y
4 15
y
5 24 2 3
2 2 (x) x dx The volume is
0 2y 3 y. The volume is
y 3) dy
y 2) dy 0 [ 2, 3] by [ 2, 3] x
2 x3 3
x.
2 2 8.
0 0 y 3). 39. A shell has radius x and height x 1 15
y
5 14
y
2 13
y
3 1
0 4
.
5 1
0 6
.
5 1. ...
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