$m\={\int }_0^{2 \mathrm{\π}}{\int }_0^{1\+\cos \left(\mathrm{\θ}\right)}r\cdot \mathrm{\ρ} \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{209}{320}\mathrm{\π}$

$I_x\={\int }_0^{2 \mathrm{\π}}{\int }_0^{1\+\cos \left(\mathrm{\θ}\right)}r\cdot \mathrm{\ρ}\cdot {\left(r \sin \left(\mathrm{\θ}\right)\right)}^2 \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{883}{2560}\mathrm{\π}$

$I_y\={\int }_0^{2 \mathrm{\π}}{\int }_0^{1\+\cos \left(\mathrm{\θ}\right)}r\cdot \mathrm{\ρ}\cdot {\left(r \cos \left(\mathrm{\θ}\right)\right)}^2 \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{2427}{2560}\mathrm{\π}$

$R_x\=\sqrt{I_y\/m}\=\sqrt{\frac{2427 \mathrm{\π}\/2560}{209 \mathrm{\π}\/320}}\=\frac{1}{836}\sqrt{1014486}$ ≐ 1.2

$R_y\=\sqrt{I_x\/m}\=\sqrt{\frac{883 \mathrm{\π}\/2560}{209 \mathrm{\π}\/320}}\=\frac{1}{836}\sqrt{369094}$ ≐ 0.73

Table 6.6.6(a)   Moments of inertia and radii of gyration

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

$\mathrm{\ρ}\=\left(1\+r^2 \left(2 {\cos }^2\left(\mathrm{\theta }\right)\+3 {\sin }^2\left(\mathrm{\theta }\right)\right)\right)\/10$$\stackrel{\text{assign}}{\to }$

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

${\int }_0^{2 \mathrm{\π}}{\int }_0^{1\+\cos \left(\mathrm{\θ}\right)}r\cdot \mathrm{\ρ} \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{209}{320}\mathrm{\π}$$\stackrel{\text{assign to a name}}{\to }$$m$

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

${\int }_0^{2 \mathrm{\π}}{\int }_0^{1\+\cos \left(\mathrm{\θ}\right)}r\cdot \mathrm{\ρ}\cdot {\left(r \sin \left(\mathrm{\θ}\right)\right)}^2 \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{883}{2560}\mathrm{\π}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Ix}$

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

${\int }_0^{2 \mathrm{\π}}{\int }_0^{1\+\cos \left(\mathrm{\θ}\right)}r\cdot \mathrm{\ρ}\cdot {\left(r \cos \left(\mathrm{\theta }\right)\right)}^2 \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{2427}{2560}\mathrm{\π}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Iy}$

Obtain $R_x$ 

$\sqrt{\mathrm{Iy}\/m}$ = $\frac{1}{836}\sqrt{1014486}$$\stackrel{\text{at 10 digits}}{\to }$$1.204804974$

Obtain $R_y$ 

$\sqrt{\mathrm{Ix}\/m}$ = $\frac{1}{836}\sqrt{369094}$$\stackrel{\text{at 10 digits}}{\to }$$0.7267118054$

Table 6.6.6(b)   Calculation of moments of inertia and radii of gyration from first principles

Total mass

$\mathrm{\ρ}≔\left(1\+r^2 \left(2 {\cos }^2\left(\mathrm{\θ}\right)\+3 {\sin }^2\left(\mathrm{\θ}\right)\right)\right)\/10\:$

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

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{1\+\cos \left(\mathrm{\θ}\right)}r\left(\frac{1}{10}\+\frac{1}{10}{r}^2\left(2{\cos \left(\mathrm{\θ}\right)}^2\+3{\sin \left(\mathrm{\θ}\right)}^2\right)\right)\ⅆr\ⅆ\mathrm{\θ}$

$m≔\mathrm{value}\left(q\right)$

$\frac{209}{320}\mathrm{\π}$

Second Moments (Moments of Inertia)

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

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{1\+\cos \left(\mathrm{\θ}\right)}{r}^3\left(\frac{1}{10}\+\frac{1}{10}{r}^2\left(2{\cos \left(\mathrm{\θ}\right)}^2\+3{\sin \left(\mathrm{\θ}\right)}^2\right)\right){\sin \left(\mathrm{\θ}\right)}^2\ⅆr\ⅆ\mathrm{\θ}$

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

$\frac{883}{2560}\mathrm{\π}$

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

${\∫}_0^{2\mathrm{\π}}{\∫}_0^{1\+\cos \left(\mathrm{\θ}\right)}{r}^3\left(\frac{1}{10}\+\frac{1}{10}{r}^2\left(2{\cos \left(\mathrm{\θ}\right)}^2\+3{\sin \left(\mathrm{\θ}\right)}^2\right)\right){\cos \left(\mathrm{\θ}\right)}^2\ⅆr\ⅆ\mathrm{\θ}$

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

$\frac{2427}{2560}\mathrm{\π}$

Radii of gyration

$$\begin{gathered} R_x\=\mathrm{sqrt}\left(\mathrm{Iy}\/m\right)\; \\ {R}_x\=\mathrm{evalf}\left(\mathrm{sqrt}\left(\mathrm{Iy}\/m\right)\right) \end{gathered}$$

$R_x\=\frac{1}{836}\sqrt{1014486}$

$$\begin{gathered} R_y\=\mathrm{sqrt}\left(\mathrm{Ix}\/m\right)\; \\ {R}_y\=\mathrm{evalf}\left(\mathrm{sqrt}\left(\mathrm{Ix}\/m\right)\right) \end{gathered}$$

$R_y\=\frac{1}{836}\sqrt{369094}$