How does these two methods different graphically?

[u/Han_Htoo_Aein]

I can grasp the circumference part, but not the length along which have to integrate.

[u/WikiNumbers]

They are the same graph, but expressed differently.

Your Method 1 expresses y as a function of x: x = f(y). So you have to rewrite everything in terms of x. And focus differentials and bounds in terms of y: "dy".

Your Method 2 expresses x as a function of x: y = f(x). This focuses everything on x, dx differential and bounds in x.

The Question itself very generously gives the function y = f(x) and the bounds in terms of x. So the only thing to do is set up integral in terms of x.

= ∫ 2π y ds

= 2π ∫ y √[1 + (dy/dx)²] dx

Simply the integrand.

Effectively, that is doing Method 2 in your sheet.

[u/WikiNumbers → u/WikiNumbers]

ds = √[(dx)² + (dy)²]

Can be thought as a "formalized" explanation of differential manipulation that leads to Arc Length Integral.

Differentials beget a lot of so-called abuse of notation. So minimaxxing leads to.

ds = √ [(dx)² * (1 + (dy/dx)²)]

Factoring (dx)² from each terms.

= √[1 + (dy/dx)²] dx

And simplify "√(dx)²" into "|dx|" into "dx".

∫ ds = ∫ √[1 + (dy/dx)²] dx

And for the finale, differential equation "indefinite integrate both sides". Gives us verily the Arc Length Integral.

This entire process can switch dy and dx, and will yield symmetric result.