Wedding ring

If you setup the integration with a parameter for the radius of the bore, then you will discover that the volume is independent of the bore radius. Therefore, set the radius to $0$ and use the known formula for the volume of a ball, $\frac{4\pi r^3}{3}$ with a value of r of 3 (one half of the bore height), $\frac{4\pi 3^3}{3}$ . The answer is $36\pi$ . Therefore the answer to be entered is $36$ .

Some will argue that since I didn't give a bore radius and a numeric answer is needed, then the bore radius is not significant and then follow the same computation.

No calculus required! Let $a$ be the radius of the ball. Then the cross section through the wedding band for a fixed value of $z$ is an annulus with radii $\sqrt{a^2-9}$ and $\sqrt{a^2-z^2}$ . The area of this annulus is $\pi(9-z^2)$ , independent of $a$ , the same as for a sphere of radius 3. By Cavalieri's Principle , the volume of the wedding band is the same as that of a sphere of radius 3, namely, $\boxed{36}\ \pi$ .

Volume of napkin ring = $\frac {π}{6}h^3$

Put h=6 to get the answer as $n=36$