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 Problem Statement

A curve y = y(x), joining two points x1 and x2 on the x axis, is revolved around
the x axis to produce a surface and a volume of revolution. Given the surface area,
find the shape of the curve y = y(x) to maximize the volume. Hint: You should
find a first integral of the Euler equation of the form yf(y, x', λ) = C. Since y = 0
at the endpoints, C = 0. Then either y = 0 for all x, or f = 0. But y ≡ 0 gives zero
volume of the solid of revolution, so for maximum volume you want to solve f = 0.
 Relevant Equations
 F+λG, Euler's Equations
My volume integral is...
$$\pi\int y^2 dx$$
My surface area integral is...
$$2\pi\int y \sqrt {1+x'^2} dy$$
I'm fairly sure the variable of integration on my volume and surface area integrals has to be the same, is that right? But when I change the variable in the surface area integral to ##2\pi\int y \sqrt {1+y'^2} dx## I'm not seeing how to get the f(y, x', λ) mentioned in the hint...
$$\pi\int y^2 dx$$
My surface area integral is...
$$2\pi\int y \sqrt {1+x'^2} dy$$
I'm fairly sure the variable of integration on my volume and surface area integrals has to be the same, is that right? But when I change the variable in the surface area integral to ##2\pi\int y \sqrt {1+y'^2} dx## I'm not seeing how to get the f(y, x', λ) mentioned in the hint...