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Given $$z = y^2 + 3,$$ give the equation of the surface if rotated around the $z$-axis.

After I plot this out, I get a simple parabola in the $yz$-plane... so flipping it about the $z$-axis is just a parabola opening down instead of opening up.. and thus I have $-z = y^2 + 3$... correct?

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    If I read the question correctly you want the equation of a surface of revolution. I don't understand the second part of your question, but you want to imagine rotating the parabola around the z axis and the surface that it sweeps out.2012-06-10
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    What you have done is reflected the graph of the parabola in the plane given by $z=0$. Rotating the parabola around the $z$-axis should yield a surface, as you say. So picturing the parabola rotated around the $z$-axis, what do you obtain?2012-06-10
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    Think polar coordinates in the $xy$ plane.2012-06-10
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    In 3space, this is just a cylinder on the zy plane correct? So if I rotate it around the z-axis... nothing changes?2012-06-10
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    Think of it this way. If you take $y = 1$ in the $yz$ plane you have the equation of a line. If you consider $(x^2+y^2)^{1/2} = 1$ then you have the equation of a cylinder in 3space. Do you see what I am hinting at?2012-06-10

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