Plan A -- the easy way to answer this problem.

The easiest solution is to start off by assuming the problem has a solution. This problem seems to be missing some critical information that you would think would be required to solve the problem, such as the radius of the sphere and the radius of the hole. But since this information wasn't supplied, you would be right to conclude that either the problem can't be solved in the manner requested (an exact real number) or else this information isn't needed. As it turns out (see below for the full scoop) this information isn't needed, so you can assume any values of the sphere's radius and the hole's radius that are consistent with a 6-inch wall of the hole. So what if you assume the hole is infinitesimally small, i.e. zero? That works, and the answer is the volume of a 6-inch diameter sphere, which is 36π.

I give full credit if you take advantage of information given away in the statement of the problem. Or to put it another way, I allow you
to assume the problem can be solved as a starting point to answering it. If you're in high school or college, this is a useful trick, which will help you on tests.

Plan B -- a slightly harder way to answer this problem.

But still no calculus is needed, well, not really...

Now, suppose you set out to answer the question in the absence of such specific direction as to how the answer should be given.  We're going to set this sphere down on an x-axis centered right down the middle of the cylindrical hole.

let x=0 be the center of the sphere, 

let r be the radius of the sphere, and

let h be the radius of the cylindrical hole.

The relationship between h and r, which we'll need later, is h² + 3² = r²

Now consider this slice of the holey sphere: Plane P is perpendicular to the axis of the hole at a directed distance of "x" from the center of the sphere, -3 ≤ x ≤ 3. The slice is the intersection of plane P with the sphere and its interior, minus the cylindrical hole. It is a pair of concentric circles, and consists of the larger circle and the points that are inside it but not inside the smaller one.

The radius of the smaller circle is h, and the radius of the larger circle is sqrt(r²-x²)

So the area of this slice is π(r²-x²)-π(h²) = π(r²-x² -h²)

Since r²=h²+3², (remember?), the area of this slice is π(h²+3²-x²-h²) = π(3²-x²)

This is the same as the area of a slice through a sphere of radius 3 with no hole, or a hole whose radius is zero.

Since the cross-section area at distance x from the center of the holey sphere is the same as the cross-section area at the same distance from the center of an ordinary sphere of radius 3, it follows that the volume of the holey sphere, which is the integral of all these cross-section areas, is the same as the volume of the ordinary sphere.

The answer is (⁴/₃)π(3³) = 36π