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Math 265 Professor Lieberman November 9, 2010 CORRECTED PRACTICE THIRD IN-CLASS EXAM DIRECTIONS: This practice is only intended to show the level of diﬃculty for Friday’s exam. The topics on the actual exam may be diﬀerent from those represented here. Also, read the directions for each problem carefully; the actual exam may only ask you to set up integrals. 1. Sketch the solid whose volume is given by the double integral ∫∫ (x2 + y 2 ) dA, R where R is the rectangle {(x, y) : 0 ≤ x ≤ 2, 0 ≤ y ≤ 3}. 2. Convert the integral ∫ 4π/3 3π/4 ∫ −5 sec θ r3 sin2 θ dr dθ 0 to rectangular coordinates and then evaluate the integral. 3. Find the center of mass of the tetrahedron bounded by the planes x + y + z = 1, x = 0, y = 0, and z = 0 if the density is proportional to the sum of the coordinates of the point. [On the in-class exam, you would only be expected to set up all the integrals for a question like this.] 2 2 4. Use spherical coordinates to ﬁnd √ the volume of the solid inside the sphere x + y + 2 z = 16, outside the cone z = x2 + y 2 and above the xy-plane.