Find the surface obtained by rotating z=x^2, y=0 about a)…
Find the surface obtained by rotating $z=x^2,\ y=0$ about a) the $z$-axis; b) the $y$-axis.
Explanation of Method
Rotating the parabola z = x² about an axis in its plane and about an axis perpendicular to it.
Step 1: a) About $Oz$
The distance from the axis, $|x_0|$, becomes $\sqrt{x^2+y^2}$, and the height $z$ stays the same: $z=x^2+y^2$.
This is a circular paraboloid.
Step 2: b) About $Oy$
The curve lies in the plane $y=0$, which is perpendicular to the $y$-axis. Every point $(x_0,0,x_0^2)$ turns on a circle in that same plane, centered at $O$, with radius $\sqrt{x_0^2+x_0^4}$. This radius takes every value in $[0,\infty)$.
\[\text{b) the whole plane } y=0\ \text{(degenerate)}\]
No proper surface of revolution arises in case b).
Result
\[\boxed{\,z=x^2+y^2\,}\]
\[\boxed{\,\text{b) the whole plane } y=0\ \text{(degenerate)}\,}\]
Remark
Correction: the source replaces the curve by $y=x^2,\ z=0$ and answers $y=x^2+z^2$. That is the rotation of a different curve. If the intended axis is $Ox$, which lies in the plane of the curve, the distance $|z_0|=x^2$ becomes $\sqrt{y^2+z^2}$:
\[ y^2+z^2=x^4 . \]