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 x3
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
 x2
3
Mass =
0 0
2
3  x2
3
Mass =
0 0
3
z 62 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
)| ] ] −
|x0 ] + [ 3
−[
x 0
3
x 3
1
10 3
1
2 x 3
3
2
[(

x

15
x
)
]
[(
4
x

36
x

108
x
)
|x0 ] + [ 9
|x0] ] −
[6 3
x 3
1
4
3
2
[ 27 [(2 x  24 x  108x  216 x) |x0] ]
Mass = (–9) + (18) – (7.5) + (12) – (6) = 7.5
** Therefore, the mass of the solid is 7.5. **