So I was wondering how I could graph 3D Volumes of Revolutions on Graphing softwares for my Investigation, but I am not sure how to do it, I have seen some youtube and geogebra links but how do I do it for a custom function? For example, a function I wanted to graph was $$y = {\sqrt{r^2 - x^2} }$$, where r is the radius. The 2D graph of this is a semicircle and when it is rotated along the x axis it forms a sphere.
Links to websites I have seen: Video I watched
Starting with $y = f(x)$, suppose you're rotating about the $x$ axis between $x=a$ and $x = b$, then the surface you want to plot is
$ (y^2 + z^2) = f^2(x) $
As an example, if $y = \sqrt{r^2 - x^2} $, where $-r \le x \le r $, then the surface is
$ y^2 + z^2 = r^2 - x^2 $ i.e. $x^2 + y^2 + z^2 = r^2 $
which is a sphere centered at the origin with radius $r$ , as expected.
Another example, $ y = a x $, where $ 0 \le x \le h $, then the surface is
$ y^2 + z^2 = a^2 x^2 $
which is the equation of a right circular cone whose axis is along the $x$ axis, and its apex at $(0,0,0)$.
You can plot both of these examples using Geogebra as shown in the linked pages.
Example 1
Example 2