How do you find the volume of these two functions?

Question

Find the volume of the region inside the surface $x^2 + y^2 + z^2 = 16$ and outside the surface $x^2 + y^2 = 4$.

How would you set this up and solve it using double integration and polar?

I came up with a graph that shows a cylinder in the middle of a sphere. I kind have the idea of subtracting the cylinder volume to the volume of the sphere but I don't know how to set it up, like what would the boundaries be?

I think I have seen the same question couple of days back ! Not sure if this is homework :P — lsp, Apr 24 '13 at 07:26
Please do not delete your question when it has gotten a good answer. Others can benefit from this thread. — robjohn, Apr 24 '13 at 17:35

score 4 · Answer 1 · edited Jun 12 '20 at 10:38

4

I am adding a picture illustrating all you need to find the volume using cylindrical coordinates. enter image description here

In fact; $$V=8\int_{r=2}^{4}\int_{\theta=0}^{\pi/2}\int_0^{\sqrt{16-r^2}}rdr d\theta dz$$

edited Jun 12 '20 at 10:38

Community

1

answered Apr 24 '13 at 07:22

Mikasa

67,374

Would I be able to solve it using double integrals? – Kenneth Hend Apr 24 '13 at 07:27
Yes. remove the third integral and evaluate $$V=4\int_{r=2}^{4}\int_{\theta=0}^{\pi/2}{\sqrt{16-r^2}}rdr d\theta dz$$ instead. – Mikasa Apr 24 '13 at 07:31
Got it. Thank you so much! – Kenneth Hend Apr 24 '13 at 07:41
You are multiplying by $4$ because of $0\leq \theta \leq 2\pi.$ – Mhenni Benghorbal Apr 24 '13 at 07:42
@MhenniBenghorbal: Oh yes. How can I make any excuses? Yes. it should have been 8 not 4. I always think positively. :-) – Mikasa Apr 24 '13 at 07:51
@FomoLoco: See the last Mhenni's remarking comment. – Mikasa Apr 24 '13 at 07:55
@amWhy: Thanks Amy. :-) and ;-) – Mikasa Apr 24 '13 at 17:57

How do you find the volume of these two functions?

1 Answers1