And I am right as well. It is not often that both parties can be right so often. Only Nelson was wrong.
You are right that it is just one orientation. When I read your essay, too quickly I admit, I read that volume was calculated in both cases but that the curve was first y=x^^3, and then x=y^^1/3.
So it is always y=x^^3 but with the curve rotated about the x axis, and integrated from 1 to infinity, not 0 to infinty as I had thought.
HOWEVER, when you solve from the volume,
you integrate pi/x from 1 to infinity,
yielding pi times the log of infinity,
which is infinite.
WHEREAS, when you solve from the surface area,
you integrate 2*pi/x^^2 from 1 to infinity,
yielding 2*pi/x from infinity to one,
which equals 2*pi.
The surface area is finite and the volume is infinite, just the opposite of what you claim Prof. Nelson got.
From the context, that your remark was that you could not paint it, but you could fill it with paint, it appears that your Professor got the wrong answer rather than it being a typo in your essay.
So you were right to imply then that if you could fill the volume with paint, you could obviously paint it. Nelson got it backwards. Volumes are always more infinite than surface areas. Your question reflects this deep intuitive insight.
But then it does not now seem necessary to change your essay.
As a former math teacher in the Lexington Mass high school, where the math team has won the national championship many times and had students on the winning US international team, I can attest to the difficulty of facing students who are smarter than yourself. In my experience as a student, it was only at Harvard that the teachers seemed smarter than the students.