080314quiz6solution

080314quiz6solution - MATH 126 Mathias Knape Last Name...

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MATH 126 QUIZ #6 Solution 03/06/08 Mathias Knape Last Name : Circle discussion section First Name : 11 am 12 pm 1 pm Note: In order to receive full credit, please show all your work (conceptual steps, deFnitions you use, . ..) 1. Use the Theorem of Pappus to ±nd the volume of the solid obtained by rotating the triangle with vertices (2,3), (2,5) and (5,4) about the x-axis. From a sketch we can see that the y -coordinate of the centroid is equal to 4. Thus the volume of the solid is V =2 π × 4 × 1 2 × 2 × 3=24 π. 2. Find the centroid of the region bounded by y = e x , y =0 , x =0and x =1 . At ±rst we ±nd the area A = R 1 0 e x dx = e x | 1 0 = e - 1. Then ¯ x = 1 A Z 1 0 xe x dx = 1 A ± xe x | 1 0 - e x | 1 0 ² = 1 e - 1 [ e - e +1] = 1 e - 1 and ¯ y = 1 A Z 1 0 1 2 e 2 x dx = 1 4 A e 2 x | 1 0 = 1 4( e - 1) ( e 2 - 1) = e +1 4 . Hence the centroid is given by ( 1 e 1 , e +1 4 ).

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080314quiz6solution - MATH 126 Mathias Knape Last Name...

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