I'm attempting to derive the maximum volume enclosed by a given surface area in a volume of revolution. I know the answer is a sphere!
As this is isoperimetric, I used lagrange multipliers for
The ds must naturally be expressed as sqrt(1 + y').dx
as the previous function was expressed in terms of dx
L = pi.y^2 + 2.pi.y.sqrt(1+y')
and have attempted to put this through the Euler-lagrange equation. The solution is hesitant to be forthcoming, despite pages of manipulation.
I have had several ideas, but none have led to progress... please help!
1) There is a constraint on the start and end points of y(x) for the volume to be enclosed, namely y(x)=0 at the endpoints.
2) people have mentioned describing the relationship between the area under y(x) and the volume of revolution, but I can't force this into a helpful form!
3) as per usual L(x,y') would be more enticing than L(y,y') although this seems improbable.
The other question is to minimise the surface area for a given volume in a volume of revolution problem, but I assume that there will be a similarity in the two answers.
I have seen so much unhelpful hinting about how to get there...
A key piece of information or a full answer would be very, very much appreciated!