Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
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. –  AnonymousCoward Jun 10 '12 at 22:05
    
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? –  Alex Petzke Jun 10 '12 at 22:06
    
Think polar coordinates in the $xy$ plane. –  AnonymousCoward Jun 10 '12 at 22:06
    
In 3space, this is just a cylinder on the zy plane correct? So if I rotate it around the z-axis... nothing changes? –  Nick Jun 10 '12 at 22:17
    
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? –  AnonymousCoward Jun 10 '12 at 22:32
add comment

1 Answer 1

$z = y^2 +3$ is a parabola opening upwards in the $yz$-plane. For any point along the $y$-axis the vertical distance from $y$ to the rotated surface is $y^2 +3$. Let $(x', y')$ be any point in the $xy$-plane. Then under rotation $(x',y')$ crosses the $y$-axis in the points where $y^2 = x'^2 + y'^2$. But as the height is constant when we rotate around the $z$-axis it follows that the surface is given by $z = x^2 + y^2 +3$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.