Lim b 9 b 2 x cos x dx lim b 2 b cos b 2 9 cos 9 2

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lim b→ ∞ 9 b 2 x cos ( x ) dx = lim b→ ∞ ( 2 b cos ( b ) 2 ( 9 ) cos ( 9 ) )= 2 ( ) cos ( ) 2 ( 9 ) cos ( 9 ) = 18 cos ( 9 ) = This integral diverges, which means that our given integral will also diverge. d. 0 x 7 + e x To determine whether this integral converges or diverges, we have to decide what comparison test we are going to use. For this integral, we are going to use the limit comparison test. We begin by saying that as x→∞ , f ( x ) looks like x 7 and then we let f ( x ) be our original function and g ( x ) be 1 x 2 . So, we can do
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lim x →∞ f ( x ) g ( x ) , which results in lim x →∞ x 9 + x 2 e x which is equal to 1 and means that the LCT applies. The integral 0 1 x 2 dx converges by the p-test with p=1. Thus the integral 0 x 7 + e x also converges by the LCT. 3. A. We are asked to compute the volume of plastic required to keep the tower printing forever. To do this we apply the shell method and find the volume of a shell which is Δ x thick. Using the shell method we know that each shell has a volume that is equal to ( 2 π rh ) Δ x . The radius of a shell will simply be x-1 because any given shell will always be a distance x away from the y axis and we subtract this distance from 1 to account for the center tower. The height of a shell
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  • Fall '07
  • Irena
  • lim, #, 3 M, LCT, 116 Sec

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