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Math380_Summer08_Exam3

# Math380_Summer08_Exam3 - Name Math 380 Exam 3 50 points...

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Name: Math 380 – Exam 3 July 30, 2008 50 points possible 1. (a) (3pts) State the conclusion to the Change of Variables Theorem for Double Integrals and explain the role of the Jacobian factor in the integrand. (b) (7pts) Determine an appropriate change of variables and evaluate the integral ZZ R e x - y x + y dA where R is the rectangle bounded by the lines y = x , y = x + 5, y = 2 - x and y = 4 - x . (More space on following page if needed.) 1. (b) Continued. 2. Consider the change of variables T ( u, v, w ) = ( u 2 + 8 uw, v, u 2 w + 3 ) . Let D * be the set of ( u, v, w ) [ - 1 , 1] × [0 , 2] × [ - 1 , 0]. (a) (2pts) Find the Jacobian of T . (b) (2pts) Are there any points in D * where this change of variables may not be appropri- ate? Briefly explain. (c) (2pts) If f ( u, v, w ) is continuous on D * and f ( T ( u, v, w )) is continuous on D = T ( D * ), explain why this change of variables is still valid and the integral will evaluate correctly. 3. (5pts) Set up but do not evaluate the following integral in cylindrical coordinates: Z 3 - 3 Z 0 - 9 - y 2 Z 3 x 2 + y 2 xz dz dx dy. 4. (a) (2pts) State the formula at the conclusion of Green’s Theorem. (b) (2pts) Let C be the boundary of the square with vertices (0 , 0), (1 , 0), (0

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