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Final-Fall-2007-2008-2

# Final-Fall-2007-2008-2 - the plane z = 0 from above by the...

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American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2007-2008 Final Exam Name: ............................... ID #: ................................. Exercise 1 (12 points) Discuss the convergence of the following series: a ) + X n =0 1 n ! b ) + X n =0 n + 10 n ln 3 n c ) + X n =0 ( - 1) n 2 1 n 2 + n Exercise 2 (10 points) If w = f ( x, y ) is differentiable and x = r + s, y = r - s , show that ∂w ∂r × ∂w ∂s = ∂f ∂x 2 - ∂f ∂y 2 Exercise 3 (15 points) Find the absolute extrema of f ( x, y ) = 5 + 4 x - 2 x 2 + 3 y - y 2 on the region R bounded by the lines y = 2 , y = x , and y = - x . Exercise 4 (8 points) Use Lagrange multipliers to find the maximum and minimum of f ( x, y ) = 4 xy subject to x 2 + y 2 = 8. Exercise 5 (15 points) Evaluate the integral Z 2 0 Z 4 - x 2 0 Z x 0 sin(2 z ) 4 - z dydzdx Exercise 6 (10 points) Evaluate the integral Z Z D cos( x 2 + y 2 ) dA , where D = { ( x, y ) R 2 ; 1 x 2 + y 2 2 , y 0 } Exercise 7 (20 points) Let V be the volume of the region D
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Unformatted text preview: the plane z = 0, from above by the sphere x 2 + y 2 + z 2 = 4 and on the sides by the cylinder x 2 + y 2 = 1 a. express V as an iterated triple integral cartesian coordinates in the order dzdydx b. express V as an iterated triple integral cartesian coordinates in the order dydzdx c. express V as an iterated triple integral spherical coordinates d. express V as an iterated triple integral cylindrical coordinates, then evaluate the resulting integral. Exercise 8 (10 points) Set up an integral in rectangular coordinates equivalent to the integral Z π/ 2 Z √ 3 1 Z √ 4-r 2 1 r 3 (sin θ cos θ ) z 2 dzdrdθ (do not evaluate the integral) good luck...
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