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

[City88]

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.

 

[gabbagabbahey]

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.