Please allow me to preface: I'm an undergraduate physics student at a small school where upper-level courses are on a two year rotation. So, I'm currently in an advanced Mathematical Methods course for which I lack prerequisites. I'm only concurrently enrolled in differential equations, I've a poor background in calculus and, I've yet to take modern physics--I've only completed the introductory level mechanics and electromagnetism courses. After a fellow student explained calculus of variations to me, briefly, I realized how startlingly wrong and almost juvenile was my attempted approach to one of the problems. What only adds to my aggravation is that, this course is a distance learning course broadcast from another small school several hundred miles away. So, given all of my previous positive experiences, I'm referring to Physics Forum with hopes of a clear, concise tutorial. For all those who may contribute, you've my deep and sincere gratitude. Thank you. 1. The problem statement, all variables and given/known data A curve y(x) of length 2a is drawn between the points (0,0) and (a,0) in such a way that the solid obtained by rotating the curve about the x-axis has the largest possible volume. Find y(1/2a) 2. Relevant equations [tex]\partial[/tex]F/[tex]\partial[/tex]y - [(d/dx) *([tex]\partial[/tex]F/[tex]\partial[/tex]y' =0 ? 3. The attempt at a solution ∫ 2piF(x) dx - I’m integrating some function, which would just be the function of the curve, so, I’d have to multiply it times two, and then pi to represent the rotation? Making it a sphere would maximize the volume so, the angle at which the curve would intersect the x axis would be 90 degrees, correct? So, if I attempt to find a function to represent the changing segments of length of the curve, and I use the formula to take the square root of (1+ y’^2), then make the substitution of y’ = ctn(theta), would my theta be 90? Or would I try to solve it parametrically? Or, would I try to solve it like the Courant and Robbins problem where you try to find the maximum path around a sphere? I would think the diameter of a would be relevant, and the curve lenght of 2a would be the value of the integral for determining a component of the y(x), and maybe I would use a Lagrane multiplier, but, I don't really understand how the variables relate to one another to know how to use a Lagrange multiplier. Thank you, again.