math20c Meesue Yoo fall 09 quiz7

math20c Meesue Yoo fall 09 quiz7 - x 2 + y 2 + z 2 = 6 and...

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Quiz 7 Solution for MATH 20C, 2009 1. For the following double integral, (a) sketch the region and (b) change the order of integration and evaluate. Z 1 0 Z 1 y sin x x dxdy Ans. (b) The domain of integration is D = { ( x,y ) | 0 x 1 , 0 y x } . Z 1 0 Z x 0 sin x x dydx = Z 1 0 ± sin x x · y ² y = x y =0 dx = Z 1 0 sin xdx = [ - cos x ] 1 0 = - cos(1) + 1 2. Find the volume of the solid in R 3 bounded by y = x 2 , x = y 2 , z = x + y + 5, and z = 0. Ans. The solid is the set of points { ( x,y,z ) | 0 z x + y + 5 ,x 2 y x, 0 x 1 } . So, Volume = Z 1 0 Z x x 2 Z x + y +5 0 1 dzdydx = Z 1 0 Z x x 2 x + y + 5 dydx = Z 1 0 ± xy + 1 2 y 2 + 5 y ² y = x y = x 2 dx = Z 1 0 x x + 1 2 x + 5 x - x 3 - 1 2 x 4 - 5 x 2 dx = ± 2 5 x 2 x + 1 4 x 2 + 10 3 x x - 1 4 x 4 - 1 10 x 5 - 5 3 x 3 ² 1 0 = 2 5 + 1 4 + 10 3 - 1 4 - 1 10 - 5 3 = 59 30 . 3. Let W be the region above the sphere
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Unformatted text preview: x 2 + y 2 + z 2 = 6 and below the paraboloid z = 4-x 2-y 2 . Write down the triple integral for the volume of W in cylindrical coordinates. (Do not evaluate). Ans. The intersection of two surfaces is x 2 + y 2 = 2 when z = 2. So, the projected down image of W on the xy-plane is the disk { ( x,y ) | x 2 + y 2 2 } . Volume of W = Z 2 Z 2 Z 4-r 2 6-r 2 1 dzrdrd....
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