quiz4 - -axis, and the ellipse 1 9 x 2 + y 2 = 1. [Putnam...

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Putnam Exam Seminar Quiz 4 Fall 2010 October 11 Problem 1. Evaluate Z 4 2 p ln(9 - x ) p ln(9 - x ) + p ln( x + 3) dx. [Putnam Exam, 1987, B1] Problem 2. Find the volume of the region of points ( x,y,z ) such that ( x 2 + y 2 + z 2 + 8) 2 36( x 2 + y 2 ) . [Putnam Exam, 2006, A1] Problem 3. Let A be the area of the region in the first quadrant bounded by the line y = 1 2 x , the x -axis, and the ellipse 1 9 x 2 + y 2 = 1. Find the positive number m such that A is equal to the area of the region in the first quadrant bounded by the line y = mx , the y
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Unformatted text preview: -axis, and the ellipse 1 9 x 2 + y 2 = 1. [Putnam Exam, 1994, A2] Problem 4. Evaluate Z x-x 3 2 + x 5 2 4-x 7 2 4 6 + 1 + x 2 2 2 + x 4 2 2 4 2 + x 6 2 2 4 2 6 2 + dx. [Putnam Exam, 1997, A3] Problem 5. Let p ( x ) = 2 + 4 x + 3 x 2 + 5 x 3 + 3 x 4 + 4 x 5 + 2 x 6 . For k with 0 < k < 5 dene I k = Z x k p ( x ) dx. For which k is I k smallest?...
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