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Final_Fall-2008-2009_Kobeissi

# Final_Fall-2008-2009_Kobeissi - American University of...

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American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2008-2009 Final Exam Exercise 1 a. (5 points) If f ( u, v, w ) is a differentiable function and if u = x - y , v = y - z , and w = z - x , show that ∂f ∂x + ∂f ∂y + ∂f ∂z = 0 b. (10 points) Use the method of Lagrange multipliers to find the maximum and minimum values of f ( x, y ) = 3 x - y + 6 on the circle x 2 + y 2 = 4 Exercise 2 (10 points) Convert to polar coordinates, then evaluate the following integral Z 2 0 Z 0 - 1 - ( y - 1) 2 xy 2 dxdy Exercise 3 (12 points) Here is the region of integration of the integral = j x 6 z = y 2 y z (0,-1,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1,-1,1) (1,-1,0) Z 1 0 Z 0 - 1 Z y 2 0 dz dy dx Rewrite the integral as an equivalent iterated integral in the other 5 orders, then evaluate one of them Exercise 4 Let V be the volume of the region D that is bounded by the paraboloid z = x 2 + y 2 , and the plane z = 2 y . a) (8 points) Express V as an iterated triple integral in cartesian coordinates in the order dz dx dy (do not evaluate the integral) . b) (10 points) Express V as an iterated triple integral in cylindrical coordinates, then evaluate the resulting integral. (you may use the result: R sin 4 x dx = - sin 3 x cos x 4 - 3 cos x sin x 8 + 3 x 8 )

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