Integrating a Function over a Paraboloid Region?

[stanford1463]

Homework Statement

Hey guys, I have one question: how can I integrate the function f(x,y,z)=x + y + z over the region between the paraboloid 4-x^2-y^2 and the xy-plane?

Homework Equations

For the paraboloid region, I used polar coordinates and found the volume of the region to be 8pi. Now, I have to find the integration of the other function x+y+z in this.

The Attempt at a Solution

Alright, I tried using a triple integral to no avail (rcos($$\theta$$) +rsin(theta) +z)r drd(theta) dz. I do not know the limits of integration (hardest part of the problem for me). Is there anyway to solve this with only a double integral? Or would I have to use cylindrical/polar whatever triple integration to solve it? Thanks..!

 

[HallsofIvy]

Homework Statement

Hey guys, I have one question: how can I integrate the function f(x,y,z)=x + y + z over the region between the paraboloid 4-x^2-y^2 and the xy-plane?

Homework Equations

For the paraboloid region, I used polar coordinates and found the volume of the region to be 8pi. Now, I have to find the integration of the other function x+y+z in this.

The Attempt at a Solution

Alright, I tried using a triple integral to no avail (rcos($$\theta$$) +rsin(theta) +z)r drd(theta) dz. I do not know the limits of integration (hardest part of the problem for me). Is there anyway to solve this with only a double integral? Or would I have to use cylindrical/polar whatever triple integration to solve it? Thanks..!

How could you find the volume if you don't know the limits of integration? If it was because you integrated
[tex]\int\int (4- x^2- y^2) dA= \int\int (4- r^2) r dr d\theta[/itex] <br /> then you should be able to see that is the same as<br /> $$\int\int\int_0^{4-x^2-y} dz dA= \int\int\int_0^{4- r^2} r dzdrd\theta$$[/tex]

 

[stanford1463]

ohh...but how is the triple integral from 0 to 4-x^2-y^2 ? I know it's the function, but graphically, I don't understand. Oh well, my homework was due 10 minutes ago and I just turned it in (leaving this question blank) lol. Thanks for the answer though!