If $\rho ( x , y , z ) = x y z$ is the volume density function of a solid enclosed by $x = z ^ { 2 } , y = x ^ { 2 } , x = 1 , y = 0$ and $z = 0$ its total mass is

A) $\frac { 1 } { 28 }$
B) $\frac { 1 } { 27 }$
C) $\frac { 1 } { 26 }$
D) $\frac { 1 } { 25 }$
E) $\frac { 1 } { 24 }$

Question

If $\rho ( x , y , z ) = x y z$ is the volume density function of a solid enclosed by $x = z ^ { 2 } , y = x ^ { 2 } , x = 1 , y = 0$ and $z = 0$ its total mass is

A) $\frac { 1 } { 28 }$
B) $\frac { 1 } { 27 }$
C) $\frac { 1 } { 26 }$
D) $\frac { 1 } { 25 }$
E) $\frac { 1 } { 24 }$

Quizplus · Accepted Answer

The total mass of the solid can be found by integrating the density function over the volume of the solid. The limits for $x$ are from 0 to 1 (since $x = 1$ and $x = z^2$ imply $z$ ranges from 0 to 1), for $y$ from 0 to $x^2$ (since $y = x^2$), and for $z$ from 0 to $\sqrt{x}$ (since $x = z^2$). The integral becomes $\int_{0}^{1} \int_{0}^{x^2} \int_{0}^{\sqrt{x}} xyz \, dz\, dy\, dx$, which evaluates to $\frac{1}{28}$.

If ρ(x,y,z)=xyz\rho ( x , y , z ) = x y zρ(x,y,z)=xyz

If $\rho ( x , y , z ) = x y z$