liggett_quizsolutions1,2

liggett_quizsolutions1,2 - y = x and y = √ x about the...

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Mathematics 31B/4 – Fall 2004 – Quiz Solutions (Note: The sketches that were required are not given below.) Quiz for Thursday, October 7 1. Use substitution to evaluate Z 13 0 dx 3 p (1 + 2 x ) 2 . Solution: Let u = 1 + 2 x,du = 2 dx . Then Z 13 0 dx 3 p (1 + 2 x ) 2 = 1 2 Z 27 1 u - 2 3 du = 3 2 u 1 3 27 1 = 3 . 2. Sketch the region between the curves y = 12 - x 2 and y = x 2 - 6, and find the area of this region. Solution: The two parabolas meet at ( ± 3 , 3). Therefore, Area = Z 3 - 3 (18 - 2 x 2 ) dx = 18 x - 2 3 x 3 3 - 3 = 72 . Quiz for Thursday, October 12 1. Sketch and find the area of the region bounded by the parabola y = x 2 , the tangent line to this parabola at (1 , 1), and the x -axis. Solution: The equation of the tangent line is y = 2 x - 1. Therefore, Area = Z 1 2 0 x 2 dx + Z 1 1 2 ( x - 1) 2 dx = 1 12 . Alternatively, one can integrate with respect to y and end up with a single integral: Area = Z 1 0 ± y + 1 2 - y dy = 1 12 . 2. Use slices to find the volume of the solid obtained by rotating the region bounded by
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Unformatted text preview: y = x and y = √ x about the line y = 1. Be sure to sketch the region. Solution: Volume = π Z 1 • (1-x ) 2-(1-√ x ) 2 ‚ dx = π 6 . Quiz for Thursday, October 14 1. Use slices to find the volume of the solid obtained by rotating the region bounded by y = x,y = 0 ,x = 2 , and x = 4 about the line x = 1. Be sure to sketch the region. Solution: Volume = π Z 2 (9-1) dy + π Z 4 2 ( 9-( y-1) 2 ) dy = 25 1 3 π. 2. Use shells to find the volume of the solid obtained by rotating the region bounded by x = 4 y 2-y 3 and x = 0 about the x-axis. Be sure to sketch the region. Solution: Volume = 2 π Z 4 y (4 y 2-y 3 ) dy = 512 5 π....
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This note was uploaded on 04/29/2010 for the course MATH 31B 31B taught by Professor Liggett during the Spring '10 term at UCLA.

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