How do I calculate the volume of a spherical slice using integration?

[tandoorichicken]

How do I find the volume of any spherical slice?

 

[HallsofIvy]

What exactly do you mean by a spherical slice? My guess would be to take two half great-circles (from pole to pole) and "cut" to the line through center and both poles- although I would call that a "wedge".

If that's what you mean, then the volume depends only on the angle between the two arcs. The volume of the entire sphere is $$\frac{4}{3}\pi R^3$$. The volume of a wedge with angle θ between the two arcs is that times $$\frac{\theta}{2\pi}= \frac{2}{3}R^3\theta$$.

 

[tandoorichicken]

a slice:
say you have a circle. then you cut straight through the circle once, and then make another parallel slice. That's what I mean by a spherical slice. Basically you end up with a frying pan like solid.

 

[HallsofIvy]

Ah. Instead of slicing through two lines of longitude, you slice along two lines of latitude. (Apparently we slice our apples differently!)

You will have to integrate to get that. Assume the sphere is centerd at (0,0,0) and has radius R. Take the two slices to be at z= z₀ and z= z₁. For each value of z between those, a cross section will be a circle centered at (0,0,z). The radius of that circle is r= √(R²- z²) and so the circle has area π(R²- z²). Taking the thickness of a thin slice to be dz, the integral becomes
π integral from z₀ to z₁ of (R²- z²) dz. Hmm, that's easier than I thought it would be.