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Sihyun Kim Problem Find the mass of the solid bounded by the xy-plane, yz-plane, xz-plane, and the plane , if the density of the solid is given by . To find the mass of the solid, break it into small cubes of volume ∆V = ∆x ∆y ∆z. Since we know that the density is constant, we can approximate the mass of each cube by δ(x,y,z) ∆x ∆y ∆z. Because we assume that ∆x, ∆y, and ∆z are infinitesimally close to 0, we are given the following triple integral to evaluate: ( x y)dzdydx Given the plane , if we assign dz as the innermost integral, we must assign the limits of the integral as followed: 0 z (6 2 x 3 y ) Likewise, if we assign dy as the middle integral, the limits of the integral must be as followed: 2 0 y ( x 2) 3 Finally, dx, the outermost integral, must be defined by the following limits: 0 x3 Thus, the equation to find the mass of the solid is the following triple integral: 2 x 2 6 2 x 3 y 3 Mass = 0 0 0 3 ( x y )dzdydx To Evaluate: 2 x2 3 Mass = 0 0 2 3 x2 3 Mass = 0 0 3 z 62 x 3 y ( xz yz ) | (2 x 2 6 x 5 xy 6 y 3 y 2 )dydx z 0 dydx 2 5 2 2 3 y 3 x 2 dx Mass = 0 (2 x y 6 xy xy 3 y y ) |y 0 2 3 2 2 3 12 x 36 x 4 x 3 12 x 2 )dx ] + [ ( )dx ] − − [ 0 ( 0 3 3 2 2 3 10 x 30 x 3 12 x 72 x 108 ( ) dx )dx ] − [ 0 ] + [ 0 ( 6 9 3 2 3 8 x 72 x 216 x 216 )dx ] [ 0 ( 27 3 Mass = Mass = 1 1 4 3 x 3 3 3 x 3 [( x 4 x ) ] [( 4 x 18 x )| ] ] − |x0 ] + [ 3 −[ x 0 3 x 3 1 10 3 1 2 x 3 3 2 [( x 15 x ) ] [( 4 x 36 x 108 x ) |x0 ] + [ 9 |x0] ] − [6 3 x 3 1 4 3 2 [ 27 [(2 x 24 x 108x 216 x) |x0] ] Mass = (–9) + (18) – (7.5) + (12) – (6) = 7.5 ** Therefore, the mass of the solid is 7.5. **