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Unformatted text preview: Test 3 sample exam If you have any questions, or think youve spotted an error / typo, please get in touch with me. Some of the problems are a little tricky, but its good practice (and better I get rid of those problems before the actual test). 1. (a) Find and classify the critical points of f ( x,y ) = x 3- 3 x + y 2 . (6 points) (b) Use your answer to part (a) to find the global minimum and max- imum of this f on the closed disk, radius 1 and center (1 , 0). (8 points) 2. Find the volume of the region above the square with vertices (1 , 1), (- 1 , 1), (1 ,- 1), (- 1 ,- 1) and below the graph of the function f ( x,y ) = 3 x 2 +6 y 2 . (5 points) 3. Evaluate Z 1 Z 1 y 3cos( x 3 ) dxdy. (8 points) 4. (a) Show that Z - Z - e- x 2- y 2 dxdy = Z - e- x 2 dx 2 . (3 points) (b) Use your answer to part (a) (and a shift to polar coordinates on the left hand side) to show that Z - e- x 2 dx = . (7 points) 5. Let R be the region in the positive octant, inside the cylinder x 2 + y 2 = 9 and below the paraboloid z = x 2 + y 2 . Using whichever coordinate system will give the simplest result, write down (a) a double integral (4 points), and (b) a triple integral (4 points) giving the volume of R . Find the volume of R . (3 points) 6. Let f ( x,y,z ) be a function of three variables. Write down the integral of f over the region in the positive octant bounded by the parabolic cylinder x = 1- y 2 and the plane z = y + 2 in the orders (a) dzdydx (4 points), and 1 (b) dxdzdy (4 points). 7. Let R be the region consisting of all points in the first octant which are at distance at most 2 from the origin. Write down triple integrals giving the volume of R in: (a) Cartesian coordinates (4 points); (b) cylindrical coordinates (4 points);...
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This note was uploaded on 09/14/2011 for the course MATH 175 taught by Professor Aldroubi during the Spring '08 term at Vanderbilt.
- Spring '08