extrema of volume of solid of revolutions

Question

Based on the definition of the volume of a solid of revolution, i wanted to apply Euler equation and find the extrema as follows: $$ v = \pi\int^{a}_{b}{y^2dx} $$ using euler equation: $$ \frac{\partial F}{\partial y}-\frac{d}{dx}(\frac{\partial F}{\partial y'})=0 $$ $$ \frac{\partial F}{\partial y} = 2ydx, \frac{d}{dx}(\frac{\partial F}{\partial y'})=0 $$ so it yields: $$ 2y\cdot dx = 0 $$ so i dont get it, what i am supposed to do afterwards? how do i integrate $2ydx$ ? does it even mean anything?

Answer 1

Now i see that i made a mistake puting the dx in the equation which is something that cant be solved and is meaningless, on the other hand the euler lagrange equations yields the solution y = 0, which is a minima. that is, because the euler lagranage equation only yields extrema.