calculus and analysis – How to compute the max/min surface area of a donut-shape solid generated by a revolved 2D circle, as the volume of the solid doesn’t change?

Let’s call, more conventional, the small and big radius r and R, where R>=r. Then the volume and area are:

V== 2 Pi^2 r^2 R 
A== 2 Pi^2 r R

You specify V= 90 Pi^2. Then A can be written as a function of only one variable, e.g. r:

A = =V/r
R == V/(2 Pi^2 r^2)

Therefore, the largest value of A of infinity is reached for r->0.

The smallest value of A is reached if r == R:

rmin = V^(1/3)/(2^(1/3) (Pi)^(2/3))

In your case for V= 90 Pi^2

rmin= 3.55689