Find the mass of the solid in the first octant bounded by the coordinate planes and the plane $x + y + z = 1$ , where the density is $\rho ( x , y , z ) = 12 x y$
A) $\frac { 1 } { 8 }$
B) $\frac { 1 } { 10 }$
C) $\frac { 1 } { 12 }$
D) $\frac { 1 } { 14 }$

Question

Quizplus · Accepted Answer

The mass of the solid can be found by integrating the density function over the volume of the solid. The solid is bounded by the plane $x + y + z = 1$ in the first octant, so we can express $z$ as $z = 1 - x - y$. The limits for $x$ and $y$ are from 0 to 1, but we must account for the plane's boundary, so $y$ varies from 0 to $1-x$. The mass $m$ is given by the triple integral $\int_{0}^{1} \int_{0}^{1-x} \int_{0}^{1-x-y} 12xy \, dz\, dy\, dx$. Integrating $12xy$ with respect to $z$ from 0 to $1-x-y$ gives $12xy(1-x-y)$. Then, integrating with respect to $y$ and $x$ over their respective limits yields the mass of the solid. After performing these integrations, the result is $\frac{1}{10}$, which corresponds to choice B.

Find the Mass of the Solid in the First Octant x+y+z=1x + y + z = 1x+y+z=1

Find the Mass of the Solid in the First Octant $x + y + z = 1$