We're learning about triple integrals and such in class.
Here's one of the problems I'm working on:
A cylindrical drill with radius 3 is used to bore a hole through the center of a sphere of radius 5. Find the volume of the ring shaped solid that remains.
Now here's what I'm thinking:
Triple integrate this: $r \,dr\, d\theta\, dz$
Bounds for $r$: $3$ to $5$
Bounds for $\theta$: $0$ to $2\pi$
Bounds for $z$ : ...$0$ to $5$? (times $2$? I mean, it's $-5$ to $5$...)
However, maybe the bounds for $r$ should depend on $z$. That would make sense... right?
I wish I knew when to have the bounds depend on another variable and such. I think this one would though, because the "radius" would shrink as you increased (or decreased) z. By how much though? When $z$ is $0$, $r$ is $5$. When $z$ is $1$, $r$ is $5$, so the new distance from the $z$-axis would be $\sqrt{1 + 5^2}$. Right...?
I'd sure appreciate some pointers! I'll respond as quickly as possible.