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?
$z = y^2 +3$ 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 $z = x^2 + y^2 +3$.