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# tut9 - | x | | y | ≤ 1 and let f R → R be continuous on...

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MATH 237 Tutorial 9 Wednesday, July 12th, 2006 Double Integrals and Polar coordinates 1. In each of the integrals, sketch the region of integration and evaluate by reversing the order of integration: a) Z 1 0 Z 1 x e y 3 dydx b) Z 3 0 Z 9 y 2 y cos( x 2 ) dxdy 2. Evaluate Z R Z x y e y dA where R is the region bounded by 0 x 1 , x 2 y x . 3. Evaluate Z R Z xy 1 + x 4 dA where R is the triangular region with vertices (0 , 0) , (0 , 1) and (1 , 1). 4. Let D xy be the region in the xy –plane which is enclosed by the lines y = 2 - x, y = 4 - x, x = 0 and y = 0. a) Find the image, D uv , in the uv –plane of D xy under the mapping ( u, v ) = F ( x, y ) = x + y 2 , x - y x + y and sketch both regions. b) Use the mapping to evaluate Z D xy Z e x - y x + y dxdy . 5. Let D xy be the region in the xy –plane defined by
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Unformatted text preview: | x | + | y | ≤ 1, and let f : R → R be continuous on the interval [-1 , 1]. Use the transformation u = x + y, v = x-y to show that Z D xy Z f ( x + y ) dxdy = Z 1-1 f ( u ) du . 6. Find the mass of the semicircular lamina occupying the region x 2 + y 2 ≤ a 2 , y ≥ where the mass density ρ ( x, y ) (mass per unit area) is proportional to the distance of the point ( x, y ) from the y –axis. Hint: Set ρ ( x, y ) = ky, y ≥ 0 where k is a constant. 7. Evaluate Z R Z (9-x 2-y 2 ) dA where R is the region x 2 + ( y-1) 2 ≤ 1....
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