Finding volume in Polar Coordinates

[PsychonautQQ]

Homework Statement

Find the volume of the wedge-shaped region contained in the cylinder x^2+y^2=9 bounded by the plane z=x and below by the xy plane

Homework Equations

The Attempt at a Solution

So it seems a common theme for me I have a hard time finding the limits of integration for the dθ term when I integrate in polar coordinates.

For this integral I am setting it up
triple integral rdzdrdθ where dr is between 0 and 3, dz is between 0 and rcosθ and dθ is between 0 and 2∏? is that correct?

 

[haruspex]

You need to be careful with this one. There's no substitute (AFAIK) for visualising the region. Pay particular attention to the fact that it says below by the XY plane. That imposes a bound on z, and hence on theta.

 

[PsychonautQQ]

does theta go from 0 to 45 since z=x it will create a 45 degree angle? dr between 0 and 3, and dz between 0 and rcos(theta)?

 

[LCKurtz]

Show us a picture of what you have for the region.

 

[haruspex]

does theta go from 0 to 45 since z=x it will create a 45 degree angle? dr between 0 and 3, and dz between 0 and rcos(theta)?

Theta is an angle in the XY plane, so is not directly related to z = x.
If 0 <= z <= x and x = r cos θ and r > 0, what is the range of possible values for theta?

 

[vanhees71]

Why should one use cylinder coordinates here to begin with?