# Triple integral with cylindrical coordinates

## Homework Statement

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.

## Homework Equations

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

## 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!

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

Mark44
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.

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 $$\sqrt{x^2+y^2}$$ ?

Mark44
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.

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

Mark44
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?

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