Tr Tripl iple e In Integra tegrals ls
Triple Integrals: ( , , ) f x y z dv V Important properties of the Triple integral: a f x y z dv , , a f x y z dv , , , V V ( ( , , ) f x y z g x y z dv ( , , )) f x y z dv ( , , ) g x y z dv ( , , ) . V V V f x y z dv ( , , ) f x y z dv ( , , ) f x y z dv ( , , ) V V V 1 2
Evaluating the triple integral in rectangular coordinates f x y z dv , , V x x y , b 2 2 f x y z dv , , f x y z dzdA , , V b x x y , 1 1
Example x y dv Evaluate the iterated integral V over the region V bounded by the planes: x 0 , y 0 , z 0 , x y z 1. Solution: 1 x y 1 1 x x y dv x y dzdydx V 0 0 0 z 1 x y 1 1 x x y z dydx 0 0 z 0 1 1 x x y 1 x y dydx 0 0
Remark: f x y z If ( , , ) 1 , then the volume of the solid V is V dv V ( , , ) , is density, then the mass of the solid V If x y z ( , , ) m x y z dv V The coordinates of the center of the solid V are given by: 1 1 x x ( , , ) x y z dv , y y ( , , ) x y z dv , c c m m V V 1 ( , , ) . z z x y z dv c m V
Example Find the volume and the coordinates of the center of gravity of : the region bounded by the parabolic cylinder 2 z 4 x and x 0 , y 0 , y 6 , z 0. Assuming the density to be constant k . Solution: 2 y 6 y 6 x 2 z 4 x x 2 2 z 4 x V dzdydx z dydx z 0 x 0 y 0 z 0 x 0 y 0 y 6 x 2 x 2 y 6 2 2 4 x dydx 4 x y dx y 0 0 0 0 x y x x 2 2 1 2 3 6 4 6 4 32 x dx x x 3 0 x 0
The mass of solid is : m x y z dxdydz , , k dxdydz V V k dxdydz kV 32 k V The coordinates of the centre of gravity : 1 k x xdzdydx xdzdydx c 32 k m V V 2 2 6 4 x 1 4 xdzdydx 32 3 0 0 0
k 1 y y dz dy dx ydzdydx c m 32 k V V 2 2 6 4 x 1 3 ydzdydx 32 0 0 0 1 k z z dz dy dx z dz dy dx c m 3 2 k V V 2 2 6 4 x 1 8 zdzdydx 32 5 0 0 0
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