Mind your inertia

This solution is an outline.

$\large{\delta =|x|=r|\cos { \theta } |.{ \quad I }_{ 0 }=\int { \int _{ D }^{ }{ { r }^{ 2 }\delta \quad rdrd\theta } } =}$

$\large{\int _{ 0 }^{ 2\pi }{ \int _{ 0 }^{ 1 }{ { r }^{ 2 }|r\cos { \theta } |rdrd\theta } } =\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \int _{ 0 }^{ 1 }{ { r }^{ 4 }\cos { \theta } drd\theta } } =4\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { 1 }{ 5 } \cos { \theta } d\theta } =\frac { 4 }{ 5 } }$