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MATH261 EXAM III FALL 2014 NAME: SI: SECTION NUMBER: You may NOT use calculators or any references. Show work to receive full credit. GOOD LUCK !!! Problem Points 1 15 2 15 3 25 4 23 5 22 Total 100 Score 1. Consider the solid bounded below by the plane z = 0, above by z = e−(x the sides by x2 + y 2 = e−2 . 2 +y 2 )/8 and on (a) (5 pts) Graph the volume. (b) (10 pts) Using a double integral, find the volume of the solid. 2. Consider the volume in the first octant bounded by the planes 3x + 4y + 16z = 12 and 3x + y = 3. (a) (5 pts) Graph the volume. (b) (10 pts) Set up the integral(s) for the volume when dV = dz dx dy. Do Not Evaluate. 3. Consider the mass with volume below by p √ bounded above by the plane z = 2, bounded 2 the sphere ρ = 1 with z ≥ 2/2, and bounded on the sides by z = x + y 2 with density δ = 1/(x2 + y 2 + z 2 ). (a) (5 pts) Graph the region in the yz plane. (b) (10 pts) Set up the integral for the mass in spherical coordinates. (c) (5 pts) Evaluate the integral. (d) (5 pts) What are x̄ and ȳ? Give a reason. u + 32 w. Verify that 4. (a) (5 pts) Consider the mapping x = 25 u + 12 w + 1, y = −1 2 u = 38 x − 18 y − 38 and w = 81 x + 58 y − 18 . Next, consider the triangle in the x, y plane with vertices (4, 1), (3, −2), and (1, 0). What are the vertices in the u, w plane? (b) (6 pts) Graph the triangular area R in the (x, y) plane and R0 in the (u, w) plane described above. The graphs must clearly show the vertices defined in the form (a, b). (c) (6 pts) Find the Jacobian for the mapping R to R0 . ZZ (d) (6 pts) Using the information from above, set up (2x+10y −2)2 (3x−y −3) dA in the new coordinate system. Do Not Evaluate. 2 R 2 5. (a) (12 pts) Show that F = hz 2 exz + y, x − z sin(yz), 2xzexz − y sin(yz)i is a conservative force without computing the potential function. R (b) (10 pts) Evaluate C F · dr where C is a path from the point (0, π, 2) to the point (π, 3, 0).