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MATH 261 EXAM III FALL 2015 NAME: CSU ID: SECTION NUMBER: Problem Points 1 18 2 24 You may NOT use calculators or any references. Show work to receive full credit. 3 18 4 18 GOOD LUCK !!! 5 22 Total 100 Potentially useful conversions: x = r cos(θ) y = r sin(θ) x = ρ sin(φ) cos(θ) y = ρ sin(φ) sin(θ) z = ρ cos(φ) r = ρ sin(φ) Score 1. (18 pts; 9 pts per part) Let R be the tetrahedron in the first octant bounded by the coordinate planes and the plane passing through (1, 0, 0), (0, 1, 0), and (0, 0, 2) with equation 2x + 2y + z = 2, as shown below. Using rectangular coordinates, set up the triple integral to find the volume of R in each of the two following variable orders, but DO NOT EVALUATE. (a) Z Z Z 1 dxdydz (b) Z Z Z 1 dzdydx z y x 2. (24 pts; 12 pts per part) Consider the region S inside the cylinder x2 + y 2 = 1, outside the sphere x2 + y 2 + z 2 = 1, and with z between 0 and 1. A view from above and a cross section (looking along the positive x axis) are shown below. z x z y y (a) Set up one or more triple integrals to integrate a function f (x, y, z) over S using cylindrical coordinates (using order dz dr dθ), but DO NOT EVALUATE. (b) Set up one or more triple integrals to integrate a function f (x, y, z) over S using spherical coordinates (using order dρ dφ dθ), but DO NOT EVALUATE. 3. (18 pts, as indicated) Consider the thin plate T consisting of the portion ofpthe unit disk in the first quadrant of the (x,y)-plane (as shown below). Let δ(x, y) = x2 + y 2 be the density of T . y x (a) (12 pts) Set up but DO NOT EVALUATE an integral to find the mass of T using polar coordinates. 3 (b) (6 pts) For this problem, the x-coordinate of the center of mass is . What is 2π the y-coordinate of the center of mass? Be sure to provide an explanation or a computation in support of your answer. 4. (18 pts, as indicated) The two parts of this problem are both about substitution but do not depend on each other. (a) (10 pts) Suppose we want to replace x and y with u and v using the substitution ( x = u2 + v 2 , y = 3uv − 2v 2 . Compute J, the determinant of the Jacobian for this substitution. (b) (8 pts) Suppose we want to integrate f (x, y) = x + y over the diamond U with boundary lines y = x + 1, y = x − 1, y = −x + 1, and y = −x − 1. Fill in the three blank spaces below (two bounds and the integrand) to write the integral of f (x, y) = x + y as an integral in u and v, using substitution x = u + v + 1, y = u − v + 2. The first and third bounding lines above have already been converted into (u, v) bounds and included below. DO NOT EVALUATE this integral. (Hint: You may use the fact that J = −2 for this substitution.) Z Z −1 dudv 0 5. (22 pts, as indicated) This problem is all about vector fields. The three parts are independent. (a) (6 pts) Determine (without searching for a potential function) whether the following vector field is conservative. Your answer should be yes or no, followed by your (brief) reasoning. F = hx2 − yx, y 2 − xy, z 2 − xyi (b) (8 pts) G = h2xy 3 z 4 , 3x2 y 2 z 4 , 4x2 y 3 z 3 + 2i is a conservative vector field. Find the potential function g(x, y, z) such that g(0, 0, 0) = 0. (c) (8 pts) h(x, y, z) = x2 + y 2 + z 2 is a potential function for conservative vector field H = h2x, 2y, 2zi. C is an unknown curve joining (0, 0, 0) to (1, 2, 3). Compute the work done in moving along C in the presence of H.