Find the volume of the solid obtained by rotating the region enclosed by the cur

Question

Find the volume of the solid obtained by rotating the region enclosed by the curves about the -axis over the interval .

(Use symbolic notation and fractions where needed.)

Explanation / Answer

Line Intersection: 9/x^2 = 10 - x^2 9 = 10x^2 - x^4 0 = - 9 + 10x^2 - x^4 0 = x^4 - 10x^2 + 9 0 = (x^2 - 1)(x^2 - 9) x^2 - 1 = 0 ====> x^2 = 1 =====> x = +/- 1 x^2 - 9 = 0 ====> x^2 = 9 =====> x = +/- 3 Disk Method: about y = 16 A (x) = p ( outer radius )^2 - p ( inner radius )^2 A (x) = p ( 16 - (9/x^2) )^2 - p ( 16 - (10 - x^2) )^2 A (x) = p [ 256 - 2 * 16 * 9/x^2 + 81/x^4 ) - ( 16 - 10 + x^2)^2 ] A (x) = p [ 256 - 288/x^2 + 81/x^4 - ( 6 + x^2)^2 ] A (x) = p [ 256 - 288/x^2 + 81/x^4 - ( 36 + 2 * 6 * x^2 + x^4) ] A (x) = p [ 256 - 288/x^2 + 81/x^4 - 36 - 12x^2 - x^4 ] A (x) = p [ 220 - 288/x^2 + 81/x^4 - 12x^2 - x^4 ] A (x) = p [ 220 - 288 * x^-2 + 81 * x^-4 - 12x^2 - x^4 ] -1 ? p [ 220 - 288 * x^-2 + 81 * x^-4 - 12x^2 - x^4 ] dx + -3 3 ? p [ 220 - 288 * x^-2 + 81 * x^-4 - 12x^2 - x^4 ] dx 1 p [ 220x - 288 * x^(-1)/(-1) + 81 * x^(-3)/(-3) - 4x^3 - (1/5) * x^5 ] p [ 220x + 288 * (1/x) - 27 * (1/x^3) - 4x^3 - (1/5) * x^5 ] . . .. .. . . .. . . . . . . ... . . . . . . . . .. . . . . .. . . .. . .. . -1 p [ 220x + 288 * (1/x) - 27 * (1/x^3) - 4x^3 - (1/5) * x^5 ] + . . .. . .. . . . . . .. .. . .. . .. . . .. .. . . . . . . .. .. . . . .-3 . . .. .. . . .. . . . . . . ... . . . . . . . . .. . . . . .. . . .. . .. ..3 p [ 220x + 288 * (1/x) - 27 * (1/x^3) - 4x^3 - (1/5) * x^5 ] . . .. . .. . . . . . .. .. . .. . .. . . .. .. . . . . . . .. .. . . . .1 p [ 220 * (-1 - -3) + 288 * [ (1/-1) - (1/-3) ] - 27 * [ (1/(-1)^3) - (1/(-3)^3) ] - 4 * [ (-1)^3 - (-3)^3 ] - (1/5) * (-1)^5 - (-3)^5 ] + p [ 220 * (3 - 1) + 288 * [ (1/3) - (1/1) ] - 27 * [ (1/3^3) - (1/1^3) ] - 4 * [ 3^3 - 1^3 ] - (1/5) * 3^5 - 1^5 ] p [ 220 * (-1 + 3) + 288 * [ -1 + (1/3) ] - 27 * [ -1 + (1/27) ] - 4 * [ -1 + 27 ] - (1/5) * [ -1 + 243 ] + p [ 220 * 2 + 288 * [ -2/3 ] - 27 * [ (1/27) - 1 ] - 4 * [ 27 - 1 ] - (1/5) * [ 243 - 1 ] p [ 220 * 2 + 288 * (-2/3) - 27 * (-26/27) - 4 * 26 - (1/5) * 242 ] + p [ 440 - 192 - 27 * [ -26/27 ] - 4 * [ 26 ] - (1/5) * [ 242 ] p [ 440 - 192 + 26 - 104 - 242/5 ] + p [ 440 - 192 + 26 - 104 - (242/5) ] p [ 440 - 192 + 26 - 104 - 242/5 + 440 - 192 + 26 - 104 - (242/5) ] p [ 880 - 384 + 52 - 208 - 484/5 ] (1216p/5) we could simply say that: 3 ? p [ 220 - 288 * x^-2 + 81 * x^-4 - 12x^2 - x^4 ] * 2 * dx 1

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