MATH 201 Exam 2, Fall 2009

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) = 3x − y + 6 on the circle x2 + y2 = 4 Exercise 2 (10 points) Convert to polar coordinates, then evaluate the following integral � 2 0 � 0 −√ 1−(y−1)2 xy2dxdy Exercise 3 (12 points) Here is the region of integration of the integral = j x 6 z=y2 y z (0,-1,1) ............... ............... (1,-1,1) (1,-1,0) � 1 0 � 0 −1 � y2 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 = x2+y2, and the plane z = 2y. 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: � sin4 x dx = −sin3 x cos x 4 − 3 cos x sin x 8 + 3x 8 ) Exercise 5 Let V be the volume of the region D that is bounded below by the xy-plane, above by the sphere x2 + y2 + z2 = 4, and on the sides by the cylinder x2 + y2 = 1. a) (7 points) Express V as an iterated triple integral in spherical coordinates in the order dρ dφ dθ (do not evaluate the integral). b) (8 points) Express V as an iterated triple integral in spherical coordinates in the order dφ dρ dθ (do not evaluate the integral). −→ Exercise 6 a. (6 points) Find the work done by the force F = xi + y2j + (y − z)k along the straight line from (0, 0, 0) to (1, 1, 1). b. (8 points) Evaluate � (1,π/2,e) (0,0,1) (ln z + ex sin y)dx + ex cos ydy + (x/z − z)dz c. Find the outward flux of the field F = (y − 2x)i + (x + y)j across the curve C in the first quadrant, bounded by the lines y = 0, y = x and x + y = 1. i. (10 points) by direct calculation ii. (6 points) by Green’s theorem good luck