MATH 115 Final: MATH 115 UPenn 115FinalAnsFall 2010 Exams
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Circle your professor"s name: dean heap jackson pizer: no calculators are allowed on this exam, please show all of your work. An answer without the reasons given will receive points. Make sure that you show all of your work: please indicate your nal answer clearly! Sketch the regions enclosed by the given curves, and nd the areas of the regions. (a) y = 6 x2, y = x2. 2 (b) 4x + y2 = 12, x = y. 2: (20 pts) using the washer method, nd the volume of the solid obtained by rotating the region bounded by the curves y = , y = 0, x = 1, x = 3 about the line y = 1. Evaluate the following integrals: (a) z sin t. T dt (b) z e4 e dx x n x (c) z t2e2tdt. Evaluate the following integrals. x + 3 dx x2 + 4x 12 (b)
