Answer
$I_x= \dfrac{M(b^2+c^2)}{12}$, $I_y= \dfrac{M(a^2+c^2)}{12}$, $I_z= \dfrac{M(a^2+b^2)}{12}$
$M=kabc$
Work Step by Step
Consider $I_x=\iiint_{E} (y^2+z^2) \rho(x,y,z) dV$
or, $=k\int_{-c/2}^{c/2} \int_{-b/2}^{b/2}\int_{-a/2}^{a/2}(y^2+z^2) dx dy dz $
or, $=k\int_{-c/2}^{c/2} \int_{-b/2}^{b/2}(ay^2+az^2) dy dz $
or, $=k \int_{-c/2}^{c/2} (\dfrac{ay^3}{3}+az^2y)_{-b/2}^{b/2} dz $
or, $=k \int_{-c/2}^{c/2} (\dfrac{ab^3}{(3)(4)}+abz^2) dz $
or, $= (k)(\dfrac{ab^3z}{12}+\dfrac{abz^3}{3})_{-c/2}^{c/2}$
or, $= \dfrac{kabc(b^2+c^2)}{12}$
Let $M=kabc$
Thus, $I_x= \dfrac{M(b^2+c^2)}{12}$
Hence, $I_x= \dfrac{M(b^2+c^2)}{12}$, $I_y= \dfrac{M(a^2+c^2)}{12}$, $I_z= \dfrac{M(a^2+b^2)}{12}$
