Answer
$I_x=I_y=I_z= \dfrac{2kL^5}{3}$
Work Step by Step
Consider $I_x=I_y=I_z=\iiint_{E} (y^2+z^2) \rho(x,y,z) dV$
or, $=k\int_{0}^L\int_{0}^{L}\int_0^L (y^2+z^2) dx dy dz $
or, $=k^L\int_{0}^{L} (y^2x+z^2x)_0^L dy dz $
or, $=(kL) \int_{0}^{L} (y^2+z^2) dy dz $
or, $=(kL) \int_{0}^{L} (L^3+z^2L) dz $
or, $= (kL)[\dfrac{zL^3}{3}+\dfrac{z^3L}{3}]_0^L$
or, $= \dfrac{2kL^5}{3}$
Hence, $I_x=I_y=I_z= \dfrac{2kL^5}{3}$
