Page 1 Math 251 copyright Joe Kahlig 15A 1 True or False F RR y x y dA x RRD D x y dA T If a b c d are constants then Z b F 4 x2 Z RR D 0 x a f x y dA is the volume under the the function f x y b Z d 2 2 Z b y c 2 Z x a y c r2 cos2 2r2 sin2 r drd 0 3 640 8 2r Z 2 Z Z2 4 1rdzdrd 0 4 r2 0 Z4 Z4 z 2 x 2 Z 5 f x y z dydxdz 0 0 0 6 Reverse the order of integration and then integrate answer Z 2 Z3 36 r Z 2 7 a rz 0 0 2 3 2 9 1 9 p r2 z 2 dzdrdtheta r 3 b note the sperical coordinates for the cone z 2 3x2 3y 2 is 6 Z 2 Z 6 Z6 0 Z6 0 4 cos sin d d d 0 12 2x 3 12 2x 3y 6 Z Z 8 1dzdydx 0 0 0 d x dx x y dydx x a 2 b f x y dxdy y c y c If a b c d are constants then Z 2 sin 2 Z Z 0 Z F d g r cos r sin rd dr 0 The integral Z 2Z g x y dydx 4 x2 0 F 2Z d f x y dydx x a Z Z y 2 dy 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 Z2 r cos Z 2 10 0 2 sin d d d 1 r2 sin dzdrdtheta 0 11 The densitypat any point is proportional to the distance from the point 1 1 x y k x 1 2 y 1 2 Z Z 2 4 y 2 k 2 p x 1 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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