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Unformatted text preview: Problem 1 Consider the region R enclosed by the curves y = x 2 and y = √ x . a) Use the slicing method (”washer” shaped slices) to find the volume of the solid obtained by rotating R around the y-axis. Solution: Note that R is contained between x = 0 and x = 1 and that on this domain √ x ≥ x 2 . Thus: Volume = π R 1 (( √ x ) 2- ( x 2 ) 2 ) dx = π R 1 ( x- x 4 ) dx = π ( x 2 2- x 5 5 ) | 1 = 3 π 10 b) Use the method of cylindrical shells to find the volume obtained by ro- tating R around the y-axis. Solution: For this method, we are integrating with respect to x , so the ob- servations that we made in part A are still relevant. Specifically: Volume = 2 π R 1 x ( √ x- x 2 ) dx = 2 π R 1 ( x 3 2- x 3 ) dx = 2 π ( 2 x 5 2 5- x 4 4 ) | 1 = 3 π 10 . Bonus) Explain any obvious relationship between the answers to a) and b). Solution: The region R is symmetirc about the line x = y . Any region with this symmetry will yield the same solid when rotated around x-axis as it will...
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- Spring '06
- dx, Tangent lines to circles, Formulacially