Solve the problem.
-Find the moment of inertia $I _ { Z }$ of the rectangular solid of density $\delta ( x , y , z ) = x y z$ defined by $0 \leq x \leq 9,0 \leq y \leq 7$, $0 \leq \mathrm { z } \leq 3$.
A) $\frac { 2,321,865 } { 8 }$
B) $\frac { 4,965,219 } { 16 }$
C) $\frac { 1,035,909 } { 8 }$
D) $\frac { 1,607,445 } { 8 }$

Question

Quizplus · Accepted Answer

The moment of inertia $I_Z$ of a solid about the z-axis can be calculated using the integral $$I_Z = \int_V \rho(x,y,z) (x^2 + y^2) dV,$$ where $\rho(x,y,z) = xyz$ is the density function and $V$ is the volume of the solid. Given the limits for $x$, $y$, and $z$, the integral becomes $$I_Z = \int_0^9 \int_0^7 \int_0^3 xyz(x^2 + y^2) dz dy dx.$$ Solving this triple integral gives $\frac{2,321,865}{8}$.

Solve the Problem IZI _ { Z }IZ​ Of the Rectangular Solid of Density

Solve the Problem $I _ { Z }$ Of the Rectangular Solid of Density