The problem from the book is (this is Calculus 2 stuff):
Find the volume common to two spheres, each with radius $r$, if the center of each sphere lies on the surface of the other sphere.
I put the center of one sphere at the origin, so its equation is $x^2 + y^2 + z^2 = r^2$. I put the center of the other sphere on the $x$-axis at $r$, so its equation is $(x-r)^2 + y^2 + z^2 = r^2$.
By looking at the solid down the $y$- or $z$-axis it looks like a football. By looking at it down the $x$-axis, it looks like a circle. So, the spheres meet along a plane as can be confirmed by setting the two equations equal to each other and simplifying until you get $x = r/2$.
So, my strategy is to integrate down the $x$-axis from 0 to $r/2$, getting the volume of the cap of one of the spheres and just doubling it, since the solid is symmetric. In other words, I want to take circular cross-sections along the $x$-axis, use the formula for the area of a circle to find their areas, and add them all up.
The problem with this is that I need to find an equation for $r$ in terms of $x$, and it has to be quadratic rather than linear, otherwise I'll end up with the volume of a cone rather than a sphere. But when I solve for, say, $y^2$ in one equation, plug it into the other one, and solve for $r$, I get something like $r = \sqrt{2 x^2}$, which is linear.