Rotational Volume of a sphere on the edge of a sphere.

Question

If you have sphere1 with raide R, and then a sphere2 with radie r with the mid point on the edge of sphere1. Calculate the volume of sphere 2 inside sphere 1 using rotational volume. R>r

Completely lost on this. $y=\sqrt{R^2-x^2}$ and $y=\sqrt{r^2-(x-R)^2}$

If you set them equal to calculate where they cross. then calculate the integral rotating around x from the little sphere from $R-r\rightarrow R-r^2/2R$ then you get a nightmare equation of r's and R's. Then take the left over part calculating the integral of the big sphere rotatating around the xaxis from $R-r^2/2R\rightarrow R$.

A better way to do this?

Answer 1

Suggestions for organizing the calculation:

1. Calculate the volume enclosed by a sphere of radius $\rho > 0$ lying to one side of a plane at distance $a$ from the center.

2. The volume sought, the intersection of two solid balls, is a union of two pieces of the type 1.

The point is, calculating the integral 1. using a notationally-simple lower limit of $a$ (instead of using the crossing point $R - r^{2}/(2R)$) usefully encapsulates the algebraic messiness.