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WINTER 2005 - MATH 52 MIDTERM February 15, 2005 Name: Student ID: Signature: Instructions: Print your name and student ID number and write your signature to indicate that you accept the honor code. During the test, you may not use notes, books, or calculators. Read each question carefully, and show all your work. Put a box around your final answer to each question. There are 8 problems. Point values are given in parentheses. You have 2 hours to answer all the questions. Question 1 2 3 4 5 6 7 8 Total Score Maximum 8 8 10 14 8 16 8 8 80 1 1. (8 points) Set up, but DO NOT EVALUATE, a triple integral in rectangular coordinates that represents the volume of the region bounded by the parabolic cylinder z = y 2 and the planes z = y + 2, x = 0 and x + z = 4. 2 2. (8 points) Use the transformation x = a · u · sin v · cos w y = b · u · sin v · sin w z = c · u · cos v to find the mass of the solid ellipsoid x2 y 2 z 2 + 2 + 2 ≤ 1, a2 b c if its density at the point (x, y, z) is given by δ(x, y, z) = 1 − 3 x2 y 2 z 2 − 2 − 2. a2 b c 3. (10 points) Find the centroid of the solid with density ρ ≡ 1 bounded by: 2 x + z 2 ≤ a2 y 2 + z 2 ≤ a2 z≥0 4 4. (14 points) (a) (6 points) Setp up the integral representing the surface area of the lamina cut from the cone z = x2 + y 2 by the cylinder x2 + y 2 = 2x. (b) (8 points) Let S be the portion of the paraboloid z = x2 + y 2 which lies between the planes z = 1 and z = 4. Parametrize S in cylindrical coordinates and setup an integral in cylindrical coordinates representing the surface area of S. 5 5. (8 points) Use integration in spherical coordinates to find the volume and centroid of the solid region that is inside the sphere ρ = 3, below the cone φ = π/3, and above the xy-plane φ = π/2. 6 6. (16 points) (a) (4 points) Sketch the region of integration in the following double integral: Z 4Z 2 √ 0 y 2 yex dx dy. x3 (b) (4 points) Evaluate the integral by changing the order of integration. (c) (6 points) Change the order of integration in the following integral in polar coordinates: Z π/2 Z a cos θ f (r, θ) dr dθ. −π/2 0 7 7. (8 points) Conpute the integral Z Z sign x2 − y 2 + 2 dA, R where R is the first quadrant portion of the disk x2 + y 2 ≤ 4 and the function sign(x) is defined by +1 if x ≥ 0 sign(x) = −1 if x < 0. 8 8. (8 points) Assume that the planar region R with density ρ ≡ 1 in the xy-plane is symmetric with respect to the line y = x. Show that I(a,−a) = I(0,0) + 2m · a2 where I(a,b) is the moment of inertia about an axis perpendicular to the xy-plane at the point (a, b). 9