I have this problem below and I have a few questions.
A sphere of radius $1$ overlaps a smaller sphere of radius $r$ in such a way that their inter-section is a circle of radius $r$. (In other words, they intersect in a great circle of the small sphere.) Find $r$ so that the volume inside the small sphere and outside the large sphere is as large as possible.
I don't need help with the problem itself, I'm just struggling to understand what it means by "inside the small sphere" and "outside the large sphere." Does "the inside" refer to the entire volume of the smaller sphere? Or does it refer to the volume exclusive to the smaller sphere in the intersection? I'm more confused about what is meant by the volume outside the large sphere. Wouldn't that be infinite? I think it's trying to refer to the volume exclusive to the larger sphere. My current interpretation is below. What do you guys think? (The unlabelled center is the volume of the intersection)
You have unwittingly drawn a clarifying Venn diagram. "Inside the small sphere" and "outside the large sphere" cannot be understood independently; the whole description is of the volume that first is inside the small sphere (so a finite volume already) and second, simultaneously is outside the large sphere.
"Inside the small sphere and outside the large sphere" refers only to the smaller shaded region in your diagram. Which is the shaded region below: