Triple integral with cylindrical coordinates

[MasterWu77]

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]

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

 

[Mark44]

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.

 

[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} ?

 

[Mark44]

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.

 

[MasterWu77]

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

 

[Mark44]

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?

 

[MasterWu77]

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