Pre-Calc Homework Solutions 343

Pre-Calc Homework Solutions 343 - Section 8.3 2x dx x2 1 b...

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53. (a) E 0 } x 2 2 x 1 dx 1 } 5 lim b E b 0 } x 2 2 x 1 dx 1 } 5 lim b 3 ln ( x 2 1 1) 4 5 lim b ln ( b 2 1 1) 5‘ Thus the integral diverges. (b) Both E 0 } x 2 2 x 1 dx 1 } and E 0 2‘ } x 2 2 x 1 dx 1 } must converge in order for E 2‘ } x 2 2 x 1 dx 1 } to converge. (c) lim b E b 2 b } x 2 2 x 1 dx 1 } 5 lim b 3 ln ( x 2 1 1) 4 5 lim b [ln ( b 2 1 1) 2 ln ( b 2 1 1)] 5 lim b 0 5 0. Note that } x 2 2 1 x 1 } is an odd function so E b 2 b } x 2 2 x 1 dx 1 } = 0. (d) Because the determination of convergence is not made using the method in part (c). In order for the integral to converge, there must be finite areas in both directions (toward and toward 2‘ ). In this case, there are infinite areas in both directions, but when one computes the integral over an interval [ 2 b , b ], there is cancellation which gives 0 as the result. 54. By symmetry, find the perimeter of one side, say for 0 # x # 1, y $ 0. y 2/3 5 1 2 x 2/3 y 5 (1 2 x 2/3 ) 3/2 } d d y x } 5 } 3 2 } (1 2 x 2/3 ) 1/2 1 2} 2 3 } x 2 1/3 2 52 x 2 1/3 (1 2 x 2/3 ) 1/2 1 } d
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.

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