Finding limits of integration for volume integral

Question

Compute the volume of solid enclosed between the surfaces $x^2+y^2=9$ and $x^2+z^2=9$

What should be the limits of the integrals ?

I am getting this

z from $-\sqrt{9-x^2}$to $\sqrt{9-x^2}$

x from $-\sqrt{9-y^2}$ to $ \sqrt{9-y^2}$

y from $0$ to $3$

Is this correct?

score 6 · Answer 1 · answered Jul 10 '13 at 12:36

6

As you noted, you are looking for a triple integration, but I am posting another parallel approach. If we consider the whole volume, we have $8$ times of the following solid thing(right on the top). This means that, I consider the first $1/8$ of $\mathbb R^3$ for working. And, I am assuming you can find out why the thin slab is really an square (Just examine the equations of both Cylinder). Moreover, we can move that piece from $x=0..x=3$ and so:

$$V_{1/8}=\int_0^3S_{\small{\text{slab}}}(x)dx=\int_0^3(9-x^2)dx$$

I think this way is easier than doing triple integrals, however, you may not like it. So just keep it for the later and let's wait for your desire solution. :-)

enter image description here

answered Jul 10 '13 at 12:36

Mikasa

67,374

That is an excellent diagram in any case !! – Aman Mittal Jul 10 '13 at 12:46
@AmanMittal: Thanks! :-) – Mikasa Jul 10 '13 at 12:59
1

@BabakS. You're an artist. – Git Gud Jul 10 '13 at 19:48
1

Artist, indeed, Babak! Thanks for sharing! – amWhy Jul 11 '13 at 00:05
@amWhy: Thanks Amy, Thanks Git. You can see the trace of the pencil there. I love this way of teaching. – Mikasa Jul 11 '13 at 01:28
1

@GitGud: Thanks for your nice words. – Mikasa Jul 11 '13 at 01:28
@BabakS.: I agree, what a beautiful illustration of the problem! +1 – Amzoti Jul 11 '13 at 19:20

rurouniwallace · Accepted Answer · 2013-07-10T19:58:43.033

1

The volume of any region $W$ can be found using triple integrals as follows:

$$V=\iiint_WdV$$

Your boundaries are almost correct. However you are limiting $y$ to the positive $y$ axis, which is only a half-sphere. The interval for $y$ should be $-3\le{y}\le{3}$.

edited Jul 10 '13 at 19:58

answered Jul 10 '13 at 19:52

rurouniwallace

6,266

I know this far, i am unable to determine the limits of integration – Aman Mittal Jul 10 '13 at 19:55
@AmanMittal See my latest edit. – rurouniwallace Jul 10 '13 at 20:01
Ok, That makes sense. Thats all i wanted. Thanks !! – Aman Mittal Jul 10 '13 at 20:03

Finding limits of integration for volume integral

2 Answers2