Finding the Volume of a Solid Below a Plane and Above a Paraboloid

[killersanta]

Homework Statement

The volume of the solid below the plane: z=2x and above the paraboloid z=x^2 + y^2.

I need help setting this one up, I can handle the evaluating.

The Attempt at a Solution

I just don't know.

 

[Char. Limit]

Drawing a picture always helps. Try looking at a few contour lines.

 

[killersanta]

That's where i am having a problem... I believe you are suppose to set Z to 0. But then X^2 + y^2 = 0... So x and y =0?

 

[Char. Limit]

That is one of the contour lines, yes. It's also one of the endpoints on your integral. Where is the domain that you integrate over? Hint: It's where z1=z2, where z1=2x and z2=x^2+y^2.

 

[killersanta]

I'm assuming after you set z1=z2. You solve for each? Setting the other one to zero. So x=2 and y = 0?

Is it:

2 y...2 y
S S 2x dxdy - S S x^2 + y^2 dxdy
0 0...0 0
?

 

[Char. Limit]

Well, consider that you're integrating over the circle (x-1)^2+y^2 = 1. Considering that, I might go into polar coordinates...

 

[killersanta]

thanks very much. I think i got it from here.

 

[Char. Limit]

No problem. Glad I could help.