a10 - x 2 6 xy 4 y 2 = 4 2 6 Let D xy be the region bounded...

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Math 237 Assignment 10 Due: Friday, Apr 1st * 1. Find the volume of the solid with height h ( x, y ) = xy and base D where D is bounded by y = x + 2 and y = x 2 . * 2. Evaluate the following integrals. a) RR D x + y 2 dA where D is the region bounded by x = y , x = - y and y = - 1. b) R 1 0 R y y sin x x dx dy . c) ZZ D xy ( x + 2 y ) 2 dA , where D xy is bounded by the ellipse x 2 + 4 xy + 5 y 2 = 5. * 3. Consider the regions D xy = { ( x, y ) | x 2 +4 xy +13 y 2 9 } and D uv = { ( u, v ) | u 2 + v 2 1 } . a) Find an invertible mapping F : R 2 R 2 that transforms D xy into D uv . Prove that your mapping F is invertible. b) The number of bacteria per unit area in D xy is given by c ( x, y ) = 10 9 π ( x 2 + 4 xy + 13 y 2 ) 2 . Use the Change of Variables Theorem to write an expression for the number of bacteria in D xy as double integral over D uv . c) Determine the number of bacteria in D xy . 4. Find the volume of the solid with height h ( x, y ) = 1 + xy and base D where D is bounded by y = x and y = x 2 . 5. Evaluate the following integrals. a) ZZ D xy 2 dA where D is the region bounded by y = x , y = 2 x and x = 3. b) Z 1 0 Z 1 x y p 1 - y 3 dy dx . c) ZZ D xy x 2 dA , where D xy
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Unformatted text preview: x 2 + 6 xy + 4 y 2 = 4. 2 6. Let D xy be the region bounded by y = 1-x , y = 2-x , y = 0 and x = y and let ( u,v ) = F ( x,y ) = ± x-y, 1 x + y ² . a) Sketch the image of D under F in the uv-plane. b) Find the Jacobian of F and show that it is never 0 on D . c) Find the mapping F-1 and the Jacobian for F-1 . d) Use the mapping F to evaluate ZZ D xy x-y x + y dA . 7. Use MAPLE to evaluate the following double integrals. (Use Maple Help to learn how to use the “int” command) a) R D xy ( x + y ) 3 dA where D xy is the triangular region with vertices (0 , 0), (1 , 1), and (2 , 0). b) R D xy y x dA where D xy is the region in the first quadrant bounded by y = 0, y = x , and x 2 + 4 y 2 = 4....
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