$m\={\int }_0^{\mathrm{\pi }}{\int }_0^{1}r\cdot \mathrm{\ρ} \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\mathrm{\π}\/3$

$I_x\={\int }_0^{\mathrm{\pi }}{\int }_0^{1}r\cdot \mathrm{\ρ}\cdot {\left(r \sin \left(\mathrm{\θ}\right)\right)}^2 \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\mathrm{\π}\/10$

$I_y\={\int }_0^{\mathrm{\pi }}{\int }_0^{1}r\cdot \mathrm{\ρ}\cdot {\left(r \cos \left(\mathrm{\θ}\right)\right)}^2 \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\mathrm{\π}\/10$

$R_x\=\sqrt{I_y\/m}\=\sqrt{\frac{\mathrm{\π}\/10}{\mathrm{\π}\/3}}\=\sqrt{3\/10}$

$R_y\=\sqrt{I_x\/m}\=\sqrt{\frac{\mathrm{\π}\/10}{\mathrm{\π}\/3}}\=\sqrt{3\/10}$

Table 6.6.5(a)   Moments of inertia and radii of gyration

Define the density $\mathrm{\ρ}$ in polar coordinates

$\mathrm{\ρ}\=r$$\stackrel{\text{assign}}{\to }$

Obtain $m$, the total mass in region $R$ 

${\int }_0^{\mathrm{\π}}{\int }_0^{1}r\cdot \mathrm{\ρ} \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{1}{3}\mathrm{\π}$$\stackrel{\text{assign to a name}}{\to }$$m$

Obtain $I_x$, the moment of inertia about the $x$-axis

${\int }_0^{\mathrm{\pi }}{\int }_0^{1}r\cdot \mathrm{\ρ}\cdot {\left(r \sin \left(\mathrm{\θ}\right)\right)}^2 \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{1}{10}\mathrm{\π}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Ix}$

Obtain $I_y$, the moment of inertia about the $y$-axis

${\int }_0^{\mathrm{\pi }}{\int }_0^{1}r\cdot \mathrm{\ρ}\cdot {\left(r \cos \left(\mathrm{\theta }\right)\right)}^2 \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{1}{10}\mathrm{\π}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Iy}$

Obtain $R_x$ 

$\sqrt{\mathrm{Iy}\/m}$ = $\frac{1}{10}\sqrt{30}$ 

Obtain $R_y$ 

$\sqrt{\mathrm{Ix}\/m}$ = $\frac{1}{10}\sqrt{30}$

Table 6.6.5(b)   Calculation of moments of inertia and radii of gyration from first principles

Total mass

$\mathrm{\ρ}≔r\:$

$q≔\mathrm{Int}\left(r\cdot \mathrm{\ρ}\,\left[r\=0..1\,\mathrm{\θ}\=0..\mathrm{\π}\right]\right)$

${\∫}_0^{\mathrm{\π}}{\∫}_0^1{r}^2\ⅆr\ⅆ\mathrm{\θ}$

$m≔\mathrm{value}\left(q\right)$

$\frac{1}{3}\mathrm{\π}$

Second Moments (Moments of Inertia)

$q≔\mathrm{Int}\left(r\cdot \mathrm{\ρ}\cdot {\left(r \sin \left(\mathrm{\θ}\right)\right)}^2\,\left[r\=0..1\,\mathrm{\θ}\=0.. \mathrm{\π}\right]\right)$

${\∫}_0^{\mathrm{\π}}{\∫}_0^1{r}^4{\sin \left(\mathrm{\θ}\right)}^2\ⅆr\ⅆ\mathrm{\θ}$

$\mathrm{Ix}≔\mathrm{value}\left(q\right)$

$\frac{1}{10}\mathrm{\π}$

$q≔\mathrm{Int}\left(r\cdot \mathrm{\ρ}\cdot {\left(r \cos \left(\mathrm{\theta }\right)\right)}^2\,\left[r\=0..1\,\mathrm{\theta }\=0.. \mathrm{\pi }\right]\right)$

${\∫}_0^{\mathrm{\π}}{\∫}_0^1{r}^4{\cos \left(\mathrm{\θ}\right)}^2\ⅆr\ⅆ\mathrm{\θ}$

$\mathrm{Iy}≔\mathrm{value}\left(q\right)$

$\frac{1}{10}\mathrm{\π}$

Radii of gyration

$R_x\=\sqrt{\mathrm{Iy}\/m}$

$R_x\=\frac{1}{10}\sqrt{30}$

$R_y\=\sqrt{\mathrm{Ix}\/m}$

$R_y\=\frac{1}{10}\sqrt{30}$