$m\={\int }_0^{2 \mathrm{\pi }}{\int }_0^{\mathrm{\φ}\left(\mathrm{\θ}\right)}r\cdot \mathrm{\rho } \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{3 \mathrm{\π}}{2}$

$I_x\={\int }_0^{2 \mathrm{\pi }}{\int }_0^{\mathrm{\φ}\left(\mathrm{\θ}\right)}r\cdot \mathrm{\ρ}\cdot {\left(r \sin \left(\mathrm{\θ}\right)\right)}^2 \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{3 \mathrm{\π}}{32}$

$I_y\={\int }_0^{2 \mathrm{\pi }}{\int }_0^{\mathrm{\φ}\left(\mathrm{\θ}\right)}r\cdot \mathrm{\ρ}\cdot {\left(r \cos \left(\mathrm{\theta }\right)\right)}^2 \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{3 \mathrm{\π}}{8}$$$

$R_x\=\sqrt{I_y\/m}\=\sqrt{\frac{3 \mathrm{\π}\/8}{3 \mathrm{\π}\/2}}\=\frac{1}{\sqrt{4}}\=1\/2$

$R_y\=\sqrt{I_x\/m}\=\sqrt{\frac{3 \mathrm{\π}\/32}{3 \mathrm{\π}\/2}}\=\frac{1}{\sqrt{16}}\=1\/4$

Table 6.6.10(a)   Moments of inertia and radii of gyration

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Calculus - Vector≻Integration≻Multiple Integration≻2-D≻Over an Ellipse

Table 6.6.10(b)   Total mass calculated with a task template that integrates over an ellipse

$\frac{1}{2}\sqrt{\frac{{\tan \left(\mathrm{\theta }\right)}^2\+1}{\frac{1}{4}\+{\tan \left(\mathrm{\theta }\right)}^2}}$

$\=\sqrt{\frac{{\sec }^2\left(\mathrm{\θ}\right)}{1\+4 {\tan }^2\left(\mathrm{\θ}\right)}}$

$\=1\/\sqrt{{\cos }^2\left(\mathrm{\θ}\right)\+4 \sin {}^2\left(\mathrm{\θ}\right)}$

$\=1\/\sqrt{{\cos }^2\left(\mathrm{\θ}\right)\+4\left(1-{\cos }^2\left(\mathrm{\θ}\right)\right)}$

$\=1\/\sqrt{4-3 {\cos }^2\left(\mathrm{\θ}\right)}$

Define the density $\mathrm{\ρ}\left(r\,\mathrm{\θ}\right)$  

$\mathrm{\ρ}\=3\+r \left(2 \cos \left(\mathrm{\θ}\right)\+3 \sin \left(\mathrm{\θ}\right)\right)$$\stackrel{\text{assign}}{\to }$

Compute the total mass $m$ 

${\int }_0^{2 \mathrm{\pi }}{\int }_0^{1\/\sqrt{4-3 {\cos }^2\left(\mathrm{\theta }\right)}}r\cdot \mathrm{\ρ} \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{3}{2}\mathrm{\π}$$\stackrel{\text{assign to a name}}{\to }$$m$$$

Obtain the moments of inertia

${\int }_0^{2 \mathrm{\π}}{\int }_0^{1\/\sqrt{4-3 {\cos }^2\left(\mathrm{\θ}\right)}}r\cdot \mathrm{\ρ}\cdot {\left(r \sin \left(\mathrm{\θ}\right)\right)}^2 \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{3}{32}\mathrm{\π}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Ix}$$$

${\int }_0^{2 \mathrm{\pi }}{\int }_0^{1\/\sqrt{4-3 {\cos }^2\left(\mathrm{\theta }\right)}}r\cdot \mathrm{\rho }\cdot {\left(r \cos \left(\mathrm{\theta }\right)\right)}^2 \mathit{\ⅆ}r \mathit{\ⅆ}\mathrm{\θ}$ = $\frac{3}{8}\mathrm{\π}$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{Iy}$$$

Obtain the radii of gyration

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

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

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

$R_y\=\frac{1}{4}$

Table 6.6.10(c)   Moments of inertial and radii of gyration from first principles

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$\mathrm{\ρ}≔3\+2 x\+3 y\:$

Total mass

$$\begin{gathered} \mathrm{MultiInt}\left(\mathrm{\ρ}\,\left[x\,y\right]\=\mathrm{Ellipse}\left(x^2\+4 y^2\=1\right)\,\mathrm{output}\=\mathrm{integral}\right)\; \\ m≔ \mathrm{MultiInt}\left(\mathrm{\ρ}\,\left[x\,y\right]\=\mathrm{Ellipse}\left(x^2\+4 y^2\=1\right)\right) \end{gathered}$$

$\frac{3}{2}\mathrm{\π}$

Moments of Inertia

$$\begin{gathered} \mathrm{MultiInt}\left(y^2\cdot \mathrm{\ρ}\,\left[x\,y\right]\=\mathrm{Ellipse}\left(x^2\+4 y^2\=1\right)\,\mathrm{output}\=\mathrm{integral}\right)\; \\ \mathrm{Ix}≔ \mathrm{MultiInt}\left(y^2\cdot \mathrm{\ρ}\,\left[x\,y\right]\=\mathrm{Ellipse}\left(x^2\+4 y^2\=1\right)\right) \end{gathered}$$

$\frac{3}{32}\mathrm{\π}$

$$\begin{gathered} \mathrm{MultiInt}\left(x^2\cdot \mathrm{\ρ}\,\left[x\,y\right]\=\mathrm{Ellipse}\left(x^2\+4 y^2\=1\right)\,\mathrm{output}\=\mathrm{integral}\right)\; \\ \mathrm{Iy}≔ \mathrm{MultiInt}\left(x^2\cdot \mathrm{\ρ}\,\left[x\,y\right]\=\mathrm{Ellipse}\left(x^2\+4 y^2\=1\right)\right) \end{gathered}$$

$\frac{3}{8}\mathrm{\π}$

Radii of gyration

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

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

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

$R_y\=\frac{1}{4}$