Business Calc Homework w answers_Part_61

Business Calc Homework w answers_Part_61 - Section 7.3 2....

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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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