Winter 2007 - Linshaw's Class - Exam 2

# Winter 2007 - Linshaw's Class - Exam 2 - Name: TA: Math...

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Name: PID: TA: Sec. No: Sec. Time: Math 20E. Midterm Exam 2 (with solutions) March 2, 2007 Turn oﬀ and put away your cell phone. You may use a calculator and one page of notes, but no books or other assistance. Read each question carefully, and answer each question completely. Show all of your work; no credit will be given for unsupported answers. Write your solutions clearly and legibly; no credit will be given for illegible solutions. If any question is not clear, ask for clariﬁcation. # Points Score 1 12 2 14 3 10 4 14 Σ 50

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1. Our goal in this problem is to ﬁnd the area of the region D bounded by the ellipse x 2 a 2 + y 2 b 2 = 1 , for real numbers a, b > 0. a. (4 points) Find a linear change of coordinates T : R 2 R 2 that maps the unit square with coordinates (0 , 0), (1 , 0), (0 , 1), (1 , 1) to the rectangle with coordinates (0 , 0), ( a, 0), (0 , b ), ( a, b ). Solution: T ( u, v ) = ( au, bv ). In other words, T ( u, v ) = ( x ( u, v ) , y ( u, v ) ) , where x ( u, v ) = au and y ( u, v ) = bv. b. (2 points) Describe the region D * in R 2 which maps to D under T . Solution: D * is the unit disk centered at the origin: D * = { ( u, v ) | u 2 + v 2 1 } . c. (6 points) Express the area of D as a double integral over D * , and compute this integral. Solution: The Jacobian of the transformation
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## This note was uploaded on 04/30/2008 for the course MATH 20E taught by Professor Enright during the Winter '07 term at UCSD.

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Winter 2007 - Linshaw's Class - Exam 2 - Name: TA: Math...

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