Practice exam for calc IV, Summer 2007 Exam 1Exam 1 will consist of roughly 100 points of similar problems to those below.No notes,texts or calculators will be allowed on the exam. You will be provided with a formula sheet.Answers to these problems will appear in a few days.1. (15 points) a) Compute1-32-x22x-1x2dydx.b) Sketch the region inR2over which the above integral is computed.c) Write the above integral as a sum of one or more integrals indxdyorder. You are notasked to compute the result!2.(15 points) In this problem you will computeR(x-y2)100dAwhereRis the regioninR2bounded byy= 0, y= 1, y=√x,andy=√x-1.a) Sketch the region inRin the plane.b) Guess a transformationTfrom (u, v) to (x, y) which will greatly simplify the integral.Hint: finding the inverse transformationT-1is often easierc) Sketch the region in (u, v) space which corresponds toRin (x, y) space.d) Compute the Jacobian ofT.
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Region, right circular cylinder, uniform mass density