Calculate Volume of Enclosed Region with Cylindrical Coordinates

AI Thread Summary
The discussion focuses on calculating the volume of the region enclosed between the surfaces z = x^2 + y^2 and z = 2x using cylindrical coordinates. Participants confirm that switching to cylindrical coordinates simplifies the problem, with limits of integration set as -π/2 ≤ θ ≤ π/2, 0 ≤ r ≤ 2cos(θ), and r^2 ≤ z ≤ 2r cos(θ). The intersection of the surfaces is identified as a circle in the xy-plane, centered at (1,0) with a radius of 1. The formula for the projection in cylindrical coordinates is established as r^2 = 2r cos(θ), leading to r = 2 cos(θ). Overall, the integration limits and approach using cylindrical coordinates are validated.
Imo
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"Find the volume of the region enclosed between the survaces z=x^2 + y^2 and z=2x"

I figured that the simplest way of doing this was to switch to a cylindrical co-ordinate system. Can someone check that the limits of integration are then
-\frac{\pi}{2}\leq \theta \leq\frac{\pi}{2}
0\leq\ r \leq 2\cos(\theta)
r^2\leq\ z \leq 2 r \cos(\theta)
(and the jacobian being r)

Thanks greatfully
 
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z= 2x is a plane and forms the top of the figure. You are correct that you should use cylindrical coordinates. But z= x2+ y2 is a paraboloid. it's interesection with z= 2x is z= 2x= x2[/wsup]+ y2 or x2- 2x+ 1 + y2= (x-1)2+ y2 = 1 which, projected down into the xy-plane is the circle with center (1,0) and radius 1. In cylindrical coordinates, x2+ y2= 2x is r2= 2rcos θ or
r= 2 cos &theta. THAT is the fomula you need.
 
Unless I'm missing something (which is entirely possible), is that not what I have?
 
Imo said:
Unless I'm missing something (which is entirely possible), is that not what I have?

Yes, I believe your limits of integration are correct.
 
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