midterm-exam2-sol

midterm-exam2-sol - Math 21B UC Davis, Winter 2010 Dan...

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Math 21B UC Davis, Winter 2010 Dan Romik Solutions to Midterm Exam 2 1 (25 points) The region bounded between the x -axis and the curve y = 1 - x 2 is revolved around the line y = - 1. Compute the volume of the resulting solid. (Note that the solid has a “hole”.) Solution. This is a solid of revolution of “washer” type, with outer radius R ( x ) = (1 - x 2 ) - ( - 1) = 2 - x 2 , and inner radius r ( x ) = 0 - ( - 1) = 1. The range for the x -coordinate is found by inter- secting the curve y = 1 - x 2 with the x -axis y = 0, which gives x = - 1 , +1. Therefore the volume is computed as V = Z 1 - 1 π ± (2 - x 2 ) 2 - 1 2 ² dx = π Z 1 - 1 ( 3 - 4 x 2 + x 4 ) dx = π ³ 3 × 2 - 4 3 x 3 ´ ´ ´ 1 - 1 + 1 5 x 5 ´ ´ ´ 1 - 1 µ = π ( 6 - 8 3 + 2 5 ) = 56 π 15 . 2 (25 points) A bucket weighing 10 pounds is pulled up to the roof of a 30-foot high building using a rope and pulley. The rope itself has a weight density of 0 . 3 lbs / ft. Compute the amount of work that is required, in lb-ft units. Solution.
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midterm-exam2-sol - Math 21B UC Davis, Winter 2010 Dan...

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