How Does a Cylinder Equation x² + y² = 2ay Represent Its Shape in 3D?

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SUMMARY

The equation of the cylinder in three-dimensional space is represented by x² + y² = 2ay, which can be rewritten in standard form as x² + (y - a)² = a². This indicates that the cylinder has a radius of 'a' and is centered at the point (0, a), extending parallel to the z-axis. Understanding this representation is crucial for visualizing the shape of the cylinder and solving related volume problems involving bounded regions, such as those with cones.

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Homework Statement



I have to find the volume of the region bounded by a cone and cylinder. (double integral style!)
I usually start off with a sketch, but I can't seem to figure out what this one quadratic surface looks like...

It's a cylinder (in 3-dimensional plane): x2+y2=2ay


The Attempt at a Solution



I know that the equation of a cylinder is just x^2+y^2=a in 3-d. I also know the equations of a hyperbolic, oblique, and parabolic cylinder...but I don't think this is one of those.
Anyone willing to help me out, or at least point me in some sort of direction!
Thanks.
 
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Try rewriting the equation into standard form:

x^2+y^2=2ay
\Rightarrow x^2+y^2-2ay=0
\Rightarrow x^2+(y-a)^2-a^2=0
\Rightarrow x^2+(y-a)^2=a^2

Which is the equation of a cylinder of radius a centered at (0,a) running parallel to the z-axis.
 

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