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2313-in-class-Nov2

# 2313-in-class-Nov2 - without evaluating the integrals Then...

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MAC 2313, Fall 2010 — In-class problems 11/2 These problems are solely for your edification, and will not be graded. You are encouraged to work on them in groups. 1 Let C 1 and C 2 be the two curves in the xy -plane given parametrically by C 1 : braceleftbigg x ( t ) = 0 , y ( t ) = 2 t - 1 , for 0 t 1; C 2 : braceleftbigg x ( t ) = - cos t, y ( t ) = sin t, for - π / 2 t π / 2 . a. Sketch these curves. b. First determine which of the following line integrals (with respect to arc length) is greater without evaluating the integrals . Then, evaluate both integrals to check your answer. integraldisplay C 1 17 ds and integraldisplay C 2 17 ds. b. First determine which of the following line integrals (with respect to arc length) is greater without evaluating the integrals . Then, evaluate both integrals to check your answer. integraldisplay C 1 x ds and integraldisplay C 2 x ds. b. First determine which of the following line integrals (with respect to arc length) is greater
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Unformatted text preview: without evaluating the integrals . Then, evaluate both integrals to check your answer. i C 1 y ds and i C 2 y ds. 2 A thin plate (lamina) of uniform density ρ covers the portion of the xy-plane lying in the ±rst quadrant and between the two circles of radii a and b , both centered at the origin. In other words, the lamina covers the points in the set { ( x, y ) : x, y ≥ 0 and a 2 ≤ x 2 + y 2 ≤ b 2 } . a. Find the coordinates ( x, y ) of the center of mass of this plate. (Using symmetry simpli±es this problem.) b. Suppose now that b = 1 (to make the algebra easier). Using your answer to the previous part, determine for which values of a the center of mass lies inside the lamina....
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