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Practice_Final - Mathematics 32B Solutions to Final...

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Mathematics 32B - Solutions to Final Examination Instructor : D. E. Weisbart September 9, 2009 NAME (please print legibly): Your University ID Number: Your Discussion Section and TA: Signature: There are ten questions on this examination. Only one double sided 3x5 note card may be used in this examination. You may not receive full credit for a correct answer if insufficient work is shown. QUESTION VALUE SCORE 1 15 2 10 3 10 4 10 5 15 6 10 7 10 8 10 9 15 10 10 TOTAL 115 1
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1. (15 points) The following integrals represent either areas or volumes of given regions. In each part, sketch the region and determine the integral without direct calculation by appealing only to formulas from classical geometry. If you calculate the integrals using calculus methods you will receive no credit. a. 2 0 x 0 d y d x. b. 2 π 0 2 1 2 0 r d z d r d θ. 2
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c. C is a curve parameterized by t (cos t, sin t ) with 0 t 2 π . Compute 1 2 C - y d x + x d y, using only Green’s theorem and elementary geometry.
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