Suppose I have a spherical segment like the one in the picture.
I want to find the infinitesimal volume of such a segment.
The angle between point A and B is $d\theta$. And the radius of the sphere is $R$.
Here, the volume is stated to be $\frac{\pi}{6}h(3a^2+3b^2+h^2)$. Now I try to express the volume with $R$ and $d\theta$ only, and I am having trouble with it. Any help would be appreciated.
Another approach for this, I guess, is using the Jacobian in spherical coordinates:
Integrating $dV$ from $\phi=0$ to $\phi=2\pi$:
$$\int_{\phi=0}^{\phi=2 \pi}r^2 \sin \theta dr d\theta d\phi$$ yeilds
$2\pi \cdot r^2 \sin \theta dr d\theta$. Is that correct?
Best Answer
This is better handled in cylindrical coordinates.
The infinitesimal volume is the area of the circular section times the infinitesimal height, $\pi r^2(z)dz$.
The radius as the function of the height is given by $r^2(z)=R^2-z^2$, then
$$V=\int_{z_a}^{z_a+h}\pi(R^2-z^2)dz=\pi\left(R^2z-\frac{z^3}3\right)\Big|_{z_a}^{z_a+h}.$$
Now, we know that
$$R^2=a^2+z_a^2=b^2+(z_a+h)^2.$$ By subtraction, $$(z_a+h)^2-z_a^2=2z_ah+h^2=a^2-b^2,$$ and
$$z_a=\frac{a^2-b^2-h^2}{2h},R^2=a^2+\left(\frac{a^2-b^2-h^2}{2h}\right)^2,$$ and the rest will follow.