Here's a tricky problem that I'm having some issues solving:
Find parametric equations for the surface obtained by rotating the curve $x=f(y)=4y^2−y^5$, $−2\le y\le2$, about the $y$-axis, and use the graph of $f$ to make a picture of the surface.
So, I have graphed the function, and it looks like this. However, given the interesting shape of the function, I'm having issues splicing it up. I believe doing it part-by-part might be the best option, but this is still quite tricky. Plus, I'm not sure if I should do cylindrical or spherical coordinates. Any help or solutions?
Thanks all — I'm new here, so I'm glad to be joining the community!
Best Answer
The system of equations defining the set of points that corresponds to your assignment is the following: $$ \left\{ \begin{array}{c} y = t \\ \sqrt{x^2 + z^2} = |f(y)| \\ -2 \le y \le 2 \end{array} \right. $$ Let's break this down. First, you "select" a certain plane in $\mathbb{R^3}$ by selecting all points that have a constant $y$. The second equation "asks" for a cylinder of radius $f(y)$ (and you can see here cylindrical coordinates may help you a lot) whose main axis is the $y$ axis. By putting the first and the second equation into the system, and plugging in the definition for $f(y)$, should get you a single equation that you can express in only two variables. In fact, this is absolutely natural to obtain in cylindrical coordinates.
What is actually going on is: you select a plane and a cylinder, and putting those together in a system gives you a circumference in $\mathbb{R}^3$, in this case centered on the $y$ axis. As you vary your $t$ variable, though, you get progressively different radii; if you could put every single circumference one next to the other you would get a continuous shape in three dimensions. And, guess what, since the radii vary exactly as your function $f(y)$, this will turn out to be the surface of revolution you're asking for.
And, of course, the restriction on $y$ still applies.