$m\={\int }_{-1}^1{\int }_0^{{\sqrt{1-x^2}}_{}}\mathrm{\ρ} \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{3}{8}\mathrm{\pi }-\frac{2}{3}$ 

$I_x\={\int }_{-1}^1{\int }_0^{{\sqrt{1-x^2}}_{}}\mathrm{\ρ}\cdot y^2 \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{11}{96}\mathrm{\pi }-\frac{4}{15}$$$

$I_y\={\int }_{-1}^1{\int }_0^{{\sqrt{1-x^2}}_{}}\mathrm{\ρ}\cdot x^2 \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{11}{96}\mathrm{\pi }-\frac{2}{15}$

$R_x\=\sqrt{\frac{I_y}{m}}\=\sqrt{\frac{\frac{11}{96}\mathrm{\pi }-\frac{2}{15}}{\frac{3}{8}\mathrm{\pi }-\frac{2}{3}}}$ = $\frac{\sqrt{55\mathrm{\pi }-64}}{2\sqrt{5}\sqrt{9\mathrm{\π}-16}}$ ≐ 0.067

$R_y\=\sqrt{\frac{I_x}{m}}\=\sqrt{\frac{\frac{11}{96}\mathrm{\pi }-\frac{4}{15}}{\frac{3}{8}\mathrm{\pi }-\frac{2}{3}}}$ = $\frac{\sqrt{55\mathrm{\π}-128}}{2\sqrt{5}\sqrt{9\mathrm{\π}-16}}$ ≐0.43

Table 6.6.2(a)   Moments of inertia and radii of gyration

Define the density $\mathrm{\ρ}$ 

$\mathrm{\ρ}\=x^2\+{\left(y-1\/2\right)}^2$$\stackrel{\text{assign}}{\to }$

Obtain the total mass $m$ 

${\int }_{-1}^1{\int }_0^{{\sqrt{1-x^2}}_{}}\mathrm{\ρ} \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{3}{8}\mathrm{\π}-\frac{2}{3}$$\stackrel{\text{assign to a name}}{\to }$$m$$$$$

Obtain the second moments

${\int }_{-1}^1{\int }_0^{{\sqrt{1-x^2}}_{}}\mathrm{\ρ}\cdot y^2 \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{11}{96}\mathrm{\π}-\frac{4}{15}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Ix}$$$$$

${\int }_{-1}^1{\int }_0^{{\sqrt{1-x^2}}_{}}\mathrm{\ρ}\cdot x^2 \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\frac{11}{96}\mathrm{\π}-\frac{2}{15}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Iy}$$$$$

Obtain the radii of gyration

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

$R_x\=\sqrt{\frac{\frac{11}{96}\mathrm{\π}-\frac{2}{15}}{\frac{3}{8}\mathrm{\π}-\frac{2}{3}}}$

$\stackrel{\text{simplify}}{\=}$

$R_x\=\frac{1}{10}\frac{\sqrt{5}\sqrt{55\mathrm{\π}-64}}{\sqrt{9\mathrm{\π}-16}}$

$\stackrel{\text{at 10 digits}}{\to }$

$R_x\=0.6656956599$

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

$R_y\=\sqrt{\frac{\frac{11}{96}\mathrm{\π}-\frac{4}{15}}{\frac{3}{8}\mathrm{\π}-\frac{2}{3}}}$

$\stackrel{\text{simplify}}{\=}$

$R_y\=\frac{1}{10}\frac{\sqrt{5}\sqrt{55\mathrm{\π}-128}}{\sqrt{9\mathrm{\π}-16}}$

$\stackrel{\text{at 10 digits}}{\to }$

$R_y\=0.4271347563$

Initialize

$\mathrm{\ρ}≔x^2\+{\left(y-1\/2\right)}^2\:$

Obtain the total mass $m$ 

$m≔\mathrm{int}\left(\mathrm{\ρ}\,\left[y\=0..\sqrt{1-x^2}\,x\=-1..1\right]\right)$

$\frac{3}{8}\mathrm{\π}-\frac{2}{3}$

Obtain the second moments

$\mathrm{Ix}≔\mathrm{int}\left(\mathrm{\ρ}\cdot y^2\,\left[y\=0..\sqrt{1-x^2}\,x\=-1..1\right]\right)$

$\frac{11}{96}\mathrm{\π}-\frac{4}{15}$

$\mathrm{Iy}≔\mathrm{int}\left(\mathrm{\ρ}\cdot x^2\,\left[y\=0..\sqrt{1-x^2}\,x\=-1..1\right]\right)$

$\frac{11}{96}\mathrm{\π}-\frac{2}{15}$

Obtain the radii of gyration

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

$R_x\=\frac{1}{10}\frac{\sqrt{5}\sqrt{55\mathrm{\π}-64}}{\sqrt{9\mathrm{\π}-16}}$

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

$R_y\=\frac{1}{10}\frac{\sqrt{5}\sqrt{55\mathrm{\π}-128}}{\sqrt{9\mathrm{\π}-16}}$