Is there a method to find the result of the rotation of a function in $\mathbb{R^2}$ to a shape in $\mathbb{R^3}$

Question

For example, if i rotate the function $f(x) = \sqrt{1-x^2}$ over the $x$ axis, i would get the shape $x^2 +y^2 +z^2 = 1$.

And if i rotate the function $g(x) = \sqrt{x} $ over the $x$ axis, i would get the shape $z^2 +y^2 = x$

Is there a general method for finding the equation of shape in $\mathbb{R^3}$ of the rotation of a function $f(x)$ in $\mathbb{R^2}$ over the $x$ or $y$ axis?

Answer 1

Note that, for any $x$, the curve in the $zy$-plane is a circle of radius $|f(x)|$. Thus, the surface equation resulting from the rotation around the $x$-axis is

$$z^2+y^2=[f(x)]^2$$

Similarly, the surface equation resulting from the rotation around the $y$-axis is

$$z^2+x^2=[f^{-1}(y)]^2$$