fitch3 - of the unit disk in R 2 Find e-x 2 y 2 29 D ∫...

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NAME__________________________________ Math 22 10:30 Test 3 You have 50 minutes to complete this test. Work 6 out of the 8 problems. Clearly CROSS OUT the two problems you choose not to work. Each question is worth 16 points, and you get four points for free, for a total of 100 points. You must show all work in order to receive full credit. If you have any questions, please come to the front and ask. 1. Let f x , y ( 29 = xy , and let R = x , y ( 29 : 0 x 1, 0 y 1 { } . Partition R into four subrectangles using the lines x = 1 2 and y = 1 2 . Let x i * , y j * ( 29 be the center of R ij for each subrectangle. Sketch the partition. Approximate the volume of the region under f and above R using a double Riemann sum. 2. Evaluate sin p x 3 ( 29 dxdy y 3 0 9 . 3. Find the volume of the solid obtained by intersecting the two paraboloids z = x 2 + y 2 and z = 1 - x 2 - y 2 . 4. Let D be the exterior
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Unformatted text preview: of the unit disk in R 2 . Find e-x 2 + y 2 ( 29 D ∫∫ dxdy . 5. Set up, but do not evaluate , an integral to find the surface area of the portion of the sphere x 2 + y 2 + z 2 = 16 between the planes z = 1 and z = 2. 6. Suppose V = dydxdz 4-x 2 ∫ 4-x 2 ∫ 4 ∫ . Sketch the solid whose volume is given by V . Be sure to label parts of your surface appropriately. 7. A sphere of radius k has volume 4 3 p k 3 . In each of the three coordinate systems, set up a triple integral that represents this volume. Do not evaluate the integrals. 8. Using the transformation u = x + y , v = y-2 x , show how the ellipse 5 x 2-2 xy + 2 y 2 = 1 in the xy-plane becomes a circle in the uv-plane. Find the area inside the ellipse by using this transformation....
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