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Practice Midterm Math 19B Spring 16.pdf - Practice Midterm...

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Practice Midterm Math 19B (Calculus 2)UCSC, Robert Boltje, Spring 2016Time:90 minutesNo calculators, cellphones, notes or other tools allowed.1.Compute the following definite integrals:(a)R1-3(x3)dx,(b)Rπ-πcos(θ),(c)R21ettdt.2.Find the area of the region bounded by the curvesy=ex,y= 2 andy= 1-x.3.Find the derivative of the functionA(x) =Rx-1et2dt.4.Compute the volume of a ball with radiusR(a) using Cavalieri’s principle,(b) using the shell method.5.Assume a rope of length 10mhas thickness 10mmat the one end andthickness 20mmat the other end and that the thickness varies gradually:
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Unformatted text preview:at distance x from the thin end the thickness is 10 mm + 1 1 , 000 x . Assume further that the rope has a density of 2000 kg/m 3 . The rope is hanging down vertically and you want to reel it in. Calculate how much work needs to be done when (a) the thinner end is on top, (b) the thicker end is on top. (Use the letter g in your answer instead of a numerical value). 6. Find an antiderivative of the function f ( x ) = x tan( x 2 ). 7. Evaluate the indefinite integral Z t √ 4-t 2 dt .