I have the following problem. I want to find the function whose revolution volume for a certain volume gives the minimum area. I have read that it is the hyperbola but I don't know how to prove it.
I have the following. For a function $f(x)$, we have that the volume of its revolution solid is
$$\int _a ^b \pi f(x) ^2 dx$$
The area will be
$$\int _a ^b 2 \pi f(x) dx + \pi f(a)^2 + \pi f(b) ^2$$
In the formula of the area we add at the end the area of the circles at the end of the solid.
If I take the quotient, I should optimize it for a minimum value, but I get stuck there. Any help is appreciated.