Test3Old - (c) (10 points) Evaluate I = ZZ R p x 2-2 xy + 5...

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MATH 231, Calculus III, Section 1, Fall 2007 Test 3 Date: Nov. 14, Wednesday Time: 10:00 am - 10:50 am Please write clearly, reduce answers to their simplest form, and box your answers. To receive full credit you must show ALL your work. Student's Name (Please print): __________________________________________ Pledge: On my honor as a student at the University of Virginia I have neither given nor received aid on this test. Signature: _______________________________ Problem Points Score 1 25 2 25 3 25 4 15 5 10 6 (Bonus) 10 Total 110/100
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Problem 1 (25 points) Find the volume of the region E bounded above by x 2 + y 2 + z 2 = 2 and below by z = x 2 + y 2 . 1
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Problem 2 (25 points) Consider the region R in R 2 bounded by x 2 - 2 xy + 5 y 2 = 1, and the transformation T given by x = u + v 2 , y = v 2 . (a) (10 points) Find, describe and sketch the region S in the uv -plane corresponding to R (via the transformation T in the sense that T : S R ). (b) (5 points) Find the Jacobian of T (Use the proper notation!).
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Unformatted text preview: (c) (10 points) Evaluate I = ZZ R p x 2-2 xy + 5 y 2 dA using the transformation T . 2 Problem 3 (25 points) Find the work done by the force eld F ( x,y ) = 3 x 2 i + (4 x + y 2 ) j on a particle that moves along the following paths: (a) (10 points) C 1 is the line segment from (1 , 0) to (0 , 1). (b) (15 points) C 2 is part of the curve x 2 + y 2 = 1 for which x 0 and y 0 (the particle moves counterclockwise). 3 Problem 4 (15 points) Find the mass of a ball given by x 2 + y 2 + z 2 9 if the density at any point, denoted by D ( x,y,z ), is proportional to its distance from the origin. Problem 5 (10 points) Using cylindrical coordinates set up, but do not evaluate the integral I = ZZZ E dV, where E is the region bounded above by x 2 + y 2 + z 2 = 4 and below by z = 2. 4 Bonus Problem 6 (10 points) Solve Problem 5 using spherical coordinates. 5...
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Test3Old - (c) (10 points) Evaluate I = ZZ R p x 2-2 xy + 5...

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