I am trying to sketch the part of $x^2+y^2=9$ which lies in the first octant between the surfaces $z=0$ and $z=\frac{x^2y}{3}$.
I understand that $x^2+y^2=9$ is a cylinder with radius three, extending along the z-axis. What's bothering me is the surface $z=\frac{x^2y}{3}$. I tried using traces to sketch the surface, but I am not getting anywhere close to the answer below:
How can I visualise and sketch this surface? Any help would be much appreciated.
One way of seeing this surface is slicing it with a plane parallel to the plane $z=0$. That means setting $z=k$ and imagining what the section should look like. In your case, \begin{equation} z=k\quad\Longrightarrow\quad x^2y=3k\quad\Longrightarrow\quad y=\frac{3k}{x^2} \end{equation} so if $k>0$ the sections seen from "above" look like the function $1/x^2$, while if $k<0$ they look like $-1/x^2$. In the first octant, as $z$ grows, it (almost) looks like an expanding right hyperbola getting away from the origin. Imagine yourself sitting at the point $(1,1,0)$ and looking towards the origin and upwards: you'd be below and inside a curved "hyperbolic" corner or chimney extending from the origin up above you and behind you.
This "corner" starts off very sharp near the origin and smooths out far away.