I wonder how many methods are available for finding the volume required by the question.
Two spheres (of radii $r$ and $a$, with $r \lt 2a$) meet in such a way that the centre of the one of radius $r$ lies on the surface of the one of radius $a$. Find the volume of the intersection
I'm not sure if one can use triple integral to evaluate the volume. Is there a method that use only single variable calculus?
On
You could use the formula for the volume of a spherical cap http://mathworld.wolfram.com/SphericalCap.html You just have to work out the measurments for the two caps
Hint Yes, the region of intersection is a solid of revolution about the line through the centers of the two spheres, and so can be written as the (single-variable) integral $$\int A(x) \,dx,$$ where $A(x)$ is the area of the cross-section of the region at the coordinate $x$ along that line.