Pre-Calc Homework Solutions 309

Pre-Calc Homework Solutions 309 - Section 7.3 66. Use...

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Unformatted text preview: Section 7.3 66. Use washer cross sections: a washer has inner radius 60. g ( y) 1 309 dx dy 1 2y 1 , and 1 2y 1 2 r dy b a2 r 2) a2 a2 y 2, outer radius R b a2 y 2, and 2 5/8 1 2y 1 1 2y dy area (R2 [(b 4 b a a y 2)2 (b a2 y 2)2] 2 5/8 y 2. The volume is y 2 dy a 2 4 3 2 2 2 3/2 y 3 1 5/8 4 b a2 4 b a a2 a2 2 y 2 dy 1 dy dx 5 16 5 2 4 b 2.997. 2 2 2 ab a). The area of a 2 61. f (x) 2 0 2x, and (2x)2 dx 53.226. 2 0 67. (a) Put the bottom of the bowl at (0, 1 4x2 dx evaluates, horizontal cross section is ( a The volume for height h is h a a 2 x2 1 2 x2 y 2)2 h a a (a2 y 2). using NINT, to dy dx (a2 4, y 12) 1 dV A dt y 2) dy a2y 1 3 y 3 h2(3a 3 h) . 62. f (x) 3 3 x ) 2 2x, and (b) For h 1 (3 2x) dx evaluates, using NINT, 2 1 and the area of a cross section is 24 . The rate of rise is 1 (0.2) 24 1 120 2 (3x 0 (52 dh dt to 63. f (x) 1.5 44.877. dy dx 1 2x x x2 m/sec. a2 (a2 x 2 and area x 2). The , and 1 2x x x2 2 1.5 68. (a) A cross section has radius r dx 2 0.5 2 0.5 2x x2 1 1 dx 1.5 A(x) r2 ( a2 x 2)2 volume is a a 2 2 dy dx 1 2 x 1 1 2 5 x 0.5 6.283 (a2 x 2) dx a2x 1 3 x 3 1 3 a 3 a a 64. f (x) 5 , and 2 a3 a3 1 3 a 3 2 1 x 1 1 5 dx 4 5 3/2 4 5 dx 1 x 4 3 a. 3 2 1 x 2 x 3 25 3/2 4 (b) A cross section has radius x A(y) 1 r 1 r2 1 y and area h 2y h y2 . The h2 2 4 3 x2 r2 1 y 2 h 9 3/2 4 49 3 51.313 2 volume is h 65. Hemisphere cross sectional area: ( R h) 2 2 A1. r2 1 0 2y h y2 dy h2 r2 y 1 2 r h. 3 y2 h y3 3h2 h 0 Right circular cylinder with cone removed cross sectional area: R2 Since A1 h2 A2 A2, the two volumes are equal by Cavalieri's theorem. Thus, volume of hemisphere volume of cylinder R3 1 3 R 3 volume of cone 2 3 R. 3 ...
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