# exam2 - Math 152 Au 11 K Kwa Name SO N0056 Signature...

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1) (20 points) A large bowl is constructed by rotating the graph of x = p 2 y + 4 y 3 , 0 y 13 around the y -axis (with dimensions measured in feet). (a) Find the volume of water in the bowl when it is filled to height h (i.e. 0 y h ) (Hint: use the disk method, not the shell method.) Solution. The volume of the water is when the bowl is filled to height h is V ( h ) = Z h 0 π p 2 y + 4 y 3 2 dy = π Z h 0 ( 2 y + 4 y 3 ) dy = π ( h 2 + h 4 ) ft 3 . (b) If the bowl contains 6 π ft 3 of water, how high is the water level in the bowl? (Hint: the relevant equation factors in the form ( h 2 - a )( h 2 + b ) for suitable numbers a and b .) Solution. By part (a), V ( h ) = 6 π . Thus h 2 + h 4 = 6 h 4 + h 2 - 6 = 0 ( h 2 - 2)( h 2 + 3) = 0 h 2 - 2 = 0 h = 2 ft . (c) Write down, but do NOT simplify or evaluate, a definite integral for computing the work needed to pump out the 6 π ft 3 of water out over the top of the bowl. (The density of water is 62.5lb/ft 3 .) Solution. The work required is W = Z 2 0 62 . 5 π p 2 y + 4 y 3 2 (13 - y ) dy = 62 . 5 π Z 2 0 ( 2 y + 4 y 3 ) (13 - y ) dy = 62 . 5 π Z 2 0 ( - 4 y 4 + 52 y 3 - 2 y 2 + 26 y ) dy = 62 . 5 π 78 - 68 2 15 ! 14056 . 4 ft-lb .
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