## Calculus of Variations

### Surface Area Example

Colin Mitchell

Rotating a
curve _{} about the *x*-axis to get a surface of revolution,
we can form the equation

_{}, _{}.

If we want to minimize this, we must find *u* that satisfies Euler's Equation:

_{}

We notice that *F*
does not have any explicit *x* in it,
so we can use the following form of Euler's Equation (Beltrami Identity):

_{}.

We then reach

_{}

Squaring both sides yields

_{}

Integrating gives

_{}

The constants are determined by the implicit conditions

_{}