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