Is the volume of this half-pipe really given by the formula $\frac\pi6h^2(3D-2h)$?

Question

I've come across a formula for the volume of a segment of sphere as given in the picture below.

The problem is Im not able to find that formula anywhere else other than in a local textbook of engineering. I doubt even if that formula exists. If that formula exists can you please mention to which particular geometric shape's volume does the formula belong to.

Answer 1

This is known as a spherical cap:

$$V=\frac{\pi h^2}{3}(3r-h)=\frac{\pi h^2}{6}(3D-2h)$$

Answer 2

Your shape is a spherical cap. Wikipedia gives the equivalent $$V=\frac {\pi h^2}3(3r-h)$$ where $r$ is the radius and is your $\frac D2$. An integral derivation is provided.

Answer 3

Hint for integration method:

Form an equation of a circle. Rotate appropriate arc around an axis. Disc method should do.

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Hint for geometry method:

Find the angle subtended by the arc of that solid. Find the volume of the cone formed with the chord of that arc as the base. Find the volume of the corresponding 'curved base' cone. Subtracted the two volumes.