$m\={\int }_{-1}^1{\int }_0^{{\sqrt{1-x^2}}_{}}1 \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\mathrm{\π}\/2$$$

$I_x\={\int }_{-1}^1{\int }_0^{{\sqrt{1-x^2}}_{}}{y}^2 \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\mathrm{\π}\/8$$$

$I_y\={\int }_{-1}^1{\int }_0^{{\sqrt{1-x^2}}_{}}{x}^2 \mathit{\ⅆ}y \mathit{\ⅆ}x$ = $\mathrm{\π}\/8$

$R_x\=\sqrt{I_y\/m}\=\sqrt{\frac{\mathrm{\π}\/8}{\mathrm{\π}\/2}}$ = $\frac{1}{2}$ 

$R_y\=\sqrt{I_x\/m}\=\sqrt{\frac{\mathrm{\π}\/8}{\mathrm{\π}\/2}}$ = $\frac{1}{2}$ $$

Table 6.6.1(a)   Moments of inertia and radii of gyration

Obtain the total mass $m$ 

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

Obtain the second moments

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

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

Obtain the radii of gyration

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

$R_x\=\frac{1}{2}$

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

$R_y\=\frac{1}{2}$

Obtain the total mass $m$ 

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

$\frac{1}{2}\mathrm{\π}$

Obtain the second moments

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

$\frac{1}{8}\mathrm{\π}$

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

$\frac{1}{8}\mathrm{\π}$

Obtain the radii of gyration

$R_x\=\mathrm{sqrt}\left(\frac{\mathrm{Iy}}{m}\right)$

$R_x\=\frac{1}{2}$

$R_y\=\mathrm{sqrt}\left(\frac{\mathrm{Ix}}{m}\right)$

$R_y\=\frac{1}{2}$