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# epe2 - f(4 x = 24 x-5 | ES n | ≤ M 4 b-a 5 180 n 4 2...

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Exam II Feb 24, 2003 MAC 2312 S Hudson Name Show all your work. Use the space provided, or leave a note. Don’t use a calculator or your own extra paper. You may use these formulae whenever needed: u a + bu du = 2 15 b 2 (3 bu - 2 a )( a + bu ) 3 / 2 + C, u du a + bu = 2 3 b 2 ( bu - 2 a ) a + bu + C 1) (15pt) Find the volume, when the region bounded by these curves is revolved around the y axis: y = x 3 , x = 0, y = 1. 2) (10pt) Find the arc length of the curve y = x 3 6 + 1 2 x from x = 1 to x = 3. 3) (10pt) A cone-shaped water reservoir is 20 ft in diameter at the top and 20 ft deep. If it is filled to a depth of 10 ft, how much work is required to pump all the water to the top? [water weighs 62 lbs/ft 3 ]. 1

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4) (5pts each) Compute the integral, and make any obvious simplifications: a) xe - x dx b) sin 2 ( x ) dx c) x 2 x - 3 dx 5) (10pts) Find the volume V of the solid generated by revolving around the y -axis the region under y = 3 x 2 - x 3 , from x = 0 to x = 3. 6) (10pts) Given the error estimate below, find the maximum possible error in using Simp- son’s rule to approximate ln(2) = 2 1 1 /t dt with n = 10 subintervals.
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Unformatted text preview: f (4) ( x ) = 24 x-5 . | ES n | ≤ M 4 ( b-a ) 5 180 n 4 2 7) (15pts) Choose ONE, and explain thoroughly: a) Explain the integral formula used in the Shell Method. Include a picture, a limit, a sum, the volume of a shell, and plenty of words in your answer. b) Prove the formula ln( ab ) = ln( a ) + ln( b ) using the deﬁnition of ln c) Use the formula for the derivative of f-1 and of ln( x ) to prove the derivative of e x is e x . 8) (15pts) Answer True or False: ln( xe 3 x ) = ln( x ) + exp(ln(3 x )) for all x > 0. log 2 3 × log 3 4 = 2 Simpson’s rule with n = 10 produces an exact answer for R 5 2 5 x 3 + 7 dx . R ln( x ) dx = x ln( x ) + x-C . The best approach to R sin 3 x cos 2 x dx is to set u = sin( x ). BONUS (5 pts; not supposed to be easy): Evaluate R √ 1 + e x dx by ﬁnding the correct u-substitution. You might try; u = e x , u = 1 + e x or u = √ 1 + e x . 3...
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epe2 - f(4 x = 24 x-5 | ES n | ≤ M 4 b-a 5 180 n 4 2...

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