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