Triple integral with cylindrical coordinates

1. Oct 27, 2009

MasterWu77

1. The problem statement, all variables and given/known data

Use cylindrical coordinates to evaluate the triple integral

$$\int$$$$\int$$$$\int$$ $$\sqrt{x^2+y^2}$$ dV in region E

where E is the solid bounded by the circular paraboloid z=9-(x^2+y^2) and the xy-plane.

2. Relevant equations

knowing that x = rcos$$\theta$$
y= rsin$$\theta$$
z=z
for a coordinates in clindrical

3. The attempt at a solution

I'm not sure how to get the bounds of integration for this problem. I know it has something to do with the paraboloid given but I am not very good at drawing such a figure. I do understand that you need to change the x and y into xcos and ysin in order to integral the problem. any help would be greatly appreciated!

2. Oct 28, 2009

lanedance

hey MaterWu77, start by transforming your integrand and bounding surface into cylindircal coordinates, things should simplify a fair bit

3. Oct 28, 2009

Staff: Mentor

Another relevant equation, and one you didn't show, is r2 = x2 + y2.

In cylindrical coordinates, dV can be represented as dz r dr d$\theta$. Take note of that factor of r.

You region E is such that 0 <= z <= 9 - (x2 + y2), 0 <= r <= 3, and 0 <= $\theta$ <= $2\pi$.

The resulting integral in cylindrical form looks like this. The question marks are placeholders that you need to fill in.

$$\int_{\theta = ?}^{?} \int_{r = ?}^? \int_{z = 0}^? ?~dz~r~dr~d\theta$$

Because of the symmetry of your integrand and the region E, you can integrate the part in the first quadrant and multiply your result by 4.

4. Oct 28, 2009

MasterWu77

ok i understand how to get the bounds of integration. would the function that i am integrating be r^2 which would come from the $$\sqrt{x^2+y^2}$$ ?

5. Oct 28, 2009

Staff: Mentor

No, it would be sqrt(r^2), which is just r for your region. There is also the r in dz r dr d(theta). Wasn't sure if you were including that one as well.

6. Oct 28, 2009

MasterWu77

yes i was including the r from the dz r dr d(theta) to get the r^2

7. Oct 28, 2009

Staff: Mentor

So this is what you have:
$$\int_{\theta = ?}^{?} \int_{r = ?}^? \int_{z = 0}^? r^2~dz~dr~d\theta$$
Do you have the limits of integration worked out?

8. Oct 28, 2009

MasterWu77

yes i do! and i have the rest of the problem worked out so thanks a lot for helping me out!