math192_hw7sp06

math192_hw7sp06 - HW7 Solutions 15.2 Areas Moments and...

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HW7 Solutions 15.2 Areas, Moments, and Centers of Mass 4. integraltext 2 0 integraltext y - y 2 - y dx dy = integraltext 2 0 ( 2 y - y 2 ) dy = bracketleftBig y 2 - y 2 3 bracketrightBig 2 0 = 4 - 8 3 = 4 3 8. integraltext 1 - 1 integraltext y 2 - 1 2 y 2 - 2 dx dy = integraltext 1 - 1 ( y 2 - 1 - 2 y 2 + 2 ) dy = integraltext 1 - 1 ( 1 - y 2 ) dy = bracketleftBig y - y 3 3 bracketrightBig 1 - 1 = 4 3 12. integraltext 2 - 1 integraltext y +2 y 2 dxdy = 9 2 21. M = integraltext 2 0 integraltext 4 - y y 2 / 2 dxdy = integraltext 2 0 parenleftBig 4 - y - y 2 2 parenrightBig dy = 14 3 ; M y = integraltext 2 0 integraltext 4 - y y 2 / 2 x dxdy = 1 2 integraltext 2 0 [ x 2 ] 4 - y y 2 / 2 dy = 1 2 integraltext 2 0 parenleftBig 16 - 8 y + y 2 - y 4 4 parenrightBig dy = 128 15 ; M x = integraltext 2 0 integraltext 4 - y y 2 / 2 y dxdy = integraltext 2 0 parenleftBig 4 y - y 2 - y 2 2 parenrightBig dy = 10 3 ¯ x = 64 35 and ¯ y = 5 7 24. M = 125 δ 6 ; M y = δ integraltext 5 0 integraltext 6 x - x 2 x x dy dx = δ integraltext 5 0 [ xy ] 6 x - x 2 x dx = δ integraltext 5 0 (5 x 2 - x 3 ) dx = 625 δ 12 ; M x = δ integraltext 5 0 integraltext 6 x - x 2 x y dy dx = δ 2 = integraltext 5 0 bracketleftbig y 2 bracketrightbig 6 x - x

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