Page 1 Math 251 copyright Joe Kahlig 15A 1 True or False RR F The integral F If a b c d are constants then T RR y x y dA y RRD D x y dA D f x y dA is the volume under the the function f x y Z b x a Z 2Z F 4 x2 4 x2 0 Z d x2 y 2 dydx 0 x2 dx 0 b Z d Z d Z b f x y dydx y c f x y dxdy y c x a 3 720 27 3r Z 2 1rdzdrd 0 9 r2 0 Z9 Z9 z 3 y 3 Z 5 f x y z dxdydz 0 0 0 6 Reverse the order of integration and then integrate answer Z 2 Z2 16 r Z 2 7 a rz 0 2 3 2 28 1 9 p r2 z 2 dzdrdtheta r 3 0 b note the sperical coordinates for the cone z 2 3x2 3y 2 is 6 Z 2 Z 6 Z4 0 Z1 2 2x Z 0 4 cos sin d d d 0 10 10x 5y Z 8 1dzdydx 0 0 0 y 2 dy 0 3r2 cos2 r sin r drd 4 d y c 0 Z Z3 Z g r cos r sin rd dr g x y dydx x a 2 b x a y c If a b c d are constants then Z 2 sin 2 Z Z 2Z Z T Z Page 2 Math 251 copyright Joe Kahlig 15A 9 The equation of the plane z 1 is cos 1 or sec Z2 Z 4 Z 2 0 sec 0 Z2 Z5 r cos Z 25 10 0 2 sin d d d 2 r2 sin dzdrdtheta 0 11 The densitypat any point is proportional to the distance from the point 1 1 x y k x 3 2 y 1 2 Z Z 2 4 y 2 k 2 p x 3 2 y 1 2 dxdy y2 12 a top part of a cone centered on the z axis z p x2 y 2 b a cylinder of radius 1 2 that is parallel to the z axis The cylinder is centered at x 1 2 on the x axis c sphere centered at the point 0 5 0 0 with radius of 0 5
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