设圆锥体的体积为 \(V\)，质量为 \(M\)，底面半径为 \(R\)，高为 \(H\)。
体积微元
\[\text dV=r\text dr\text dh\text d\theta
\]
体积
\[\begin{aligned}
V =& \int_0^{2\pi}\int_0^H\int_0^{r(h)}r\text dr\text dh\text d\theta \\
=& 2\pi\int_0^H\int_0^{\frac RH(H-h)}r\text dr\text dh \\
=& 2\pi\int_0^H\frac 12\left(\frac RH(H-h)\right)^2\text dh \\
\xlongequal[h=Hx]{x=h/H} & \pi R^2\int_0^1(1-2x+x^2)H\text dx \\
=& \pi R^2H\left[x-x^2+\frac{x^3}{3}\right]_0^1 \\
=& \frac 13\pi R^2H
\end{aligned}\]
质心高度
\[\begin{aligned}
z =& \frac{\int_0^{2\pi}\int_0^{H}\int_0^{r(h)}hr\text dr\text dh\text d\theta}
{\int_0^{2\pi}\int_0^{H}\int_0^{r(h)}r\text dr\text dh\text d\theta} \\
=& \frac 1{\frac 13\pi R^2H}
2\pi\int_0^Hh\int_0^{\frac RH(H-h)}r\text dr\text dh \\
=& \frac 6{R^2H}\int_0^H\frac h2\left(\frac RH(H-h)\right)^2\text dh \\
=& \frac 3H\int_0^H\frac {(H-h)^2}{H^2}h\text dh \\
\xlongequal[h=Hx]{x=h/H}& \frac 3H\int_0^1(1-2x+x^2)H^2x\text dx \\
=& 3H\left[\frac{x^2}2-\frac 23 x^3+\frac{x^4}4\right]_0^1 \\
=& \frac H4
\end{aligned}\]
质量微元
\[\text dm=\rho\text dV=\frac M{\frac 13\pi R^2H}r\text dr\text dh\text d\theta
\]
绕z轴转动惯量
\[\begin{aligned}
J_z =& \int_0^{2\pi}\int_0^H\int_0^{r(h)}r^2\text dm \\
=& \frac M{\frac 13\pi R^2H}\int_0^{2\pi}\int_0^H\int_0^{r(h)}r^2
r\text dr\text dh\text d\theta \\
=& \frac {6M}{R^2H}\int_0^H\int_0^{\frac RH(H-h)}r^3\text dr\text dh \\
=& \frac {6MR^4}{4R^2H}\int_0^H\frac{(H-h)^4}{H^4}\text dh \\
\xlongequal[h=Hx]{x=h/H}& \frac {3MR^2}{2H}\int_0^1(1-x)^4H\text dx \\
=& \frac {3MR^2}2(1-4\frac 12+6\frac 13-4\frac 14+1\frac 15) \\
=& \frac 3{10}MR^2
\end{aligned}\]
计算绕x轴转动惯量时，原点放在质心，其内一点到x轴坐标为
\[d(r,h,\theta)=\sqrt{h^2+r^2\cos^2\theta}
\]
\[\begin{aligned}
J_x=& \int_0^{2\pi}\int_0^H\int_0^{r(h)}d^2(r,h,\theta)\text dm \\
=& \frac M{\frac 13\pi R^2H}\int_0^{2\pi}\int_0^H\int_0^{\frac RH(H-h)}
(h^2+r^2\cos^2\theta)r\text dr\text dh\text d\theta \\
=& \frac M{\frac 13\pi R^2H}\left(\pi R^2\int_0^Hh^2\frac{(H-h)^2}{H^2}\text dh
+\int_0^{2\pi}\cos^2\theta\text d\theta\int_0^H
\int_0^{\frac RH(H-h)}r^3\text dr\text dh\right) \\
=& \frac M{\frac 13\pi R^2H}\left(\pi R^2\int_0^1(1-x)^2H^3x^2\text dx
+\int_0^{2\pi}\frac{\cos 2\theta+1}{2}\text d\theta\frac{R^4}4\int_0^H
\frac{(H-h)^4}{H^4}\text dh\right) \\
=& \frac M{\frac 13\pi R^2H}\left(\pi R^2H^3
\left[\frac{x^3}3-\frac{x^4}2+\frac{x^5}5\right]_0^1
+\frac{\pi R^4}4\int_0^1(1-x)^4H\text dx\right) \\
=& \frac M{\frac 13\pi R^2H}\left(\pi R^2H^3\frac 1{30}
+\frac 1{20}\pi HR^4\right) \\
=& 3M\left(\frac{H^2}{30}+\frac{R^2}{20}\right)
\end{aligned}\]