MHB Surface of Revolution: Find Equation & Identify Surface

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The equation of the surface of revolution formed by revolving the curve defined by y^2 + z^2 + 2y = 0 around the y-axis is identified as a sphere. This curve can be rewritten as (y + 1)^2 + z^2 = 1, representing a circle centered at (0, -1, 0) with a radius of 1. When this circle is revolved about the y-axis, it generates a sphere. The resulting equation for the sphere is x^2 + (y + 1)^2 + z^2 = 1. Thus, the discussion effectively identifies the surface of revolution as a sphere.
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I quote a question from Yahoo! Answers

Find the equation of the surface of revolution when y^2+z^2+2y=0 is revolved about y-axis. Identify the surface of revolution.

I have given a link to the topic there so the OP can see my response.
 
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We can express $y^2+z^2+2y=0 $ as $(y+1)^2+z^2=1$, so $\gamma\equiv y^2+z^2+2y=0,x=0$ is a circle with center $(0,-1,0)$, radius $1$, and revolving about one of its diameters. As a consquence, the corresponding surface is the sphere $E\equiv x^2+(y+1)^2+z^2=1$.
 
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