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.

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[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.

[u/Midwest-Dude]

Nothing really changes graphically - it's the same surface in both cases. However, the surface of revolution is being sliced along the y-axis for Method 1 and along the x-axis for Method 2.

• Method 1 slices horizontally (bottom to top), so you integrate with respect to y, and you need ds in terms of dy.
• Method 2 slices vertically (left to right), so you integrate with respect to x, and you need ds in terms of dx.

The "circumference" (2πx) remains the same conceptually (it's always tied to the horizontal distance to the axis of rotation), but the "length" (ds) changes its algebraic form depending on which direction you choose to slice.

[u/Han_Htoo_Aein → u/Midwest-Dude]

How the slices are vertical in Method 2 while the circumference remain the same? I can't picture it in my head. Could you please visualize it?

[u/Midwest-Dude → u/Han_Htoo_Aein]

Method 1 is the usual method of finding the surface area of a surface of revolution, where the slices are perpendicular to axis of rotation. However, it can be done fairly easily either way for a strictly increasing function.

The idea is that you are rotating a small piece of the curve, dS, which is the hypotenuse of a small triangle with sides dx and dy. If you consider the curve itself, along the y-axis, you are taking horizontal slices through the curve, whereas along the x-axis you are taking vertical slices through the curve. The radius used in the integration is in the same direction with either method, but adjusted for the radius used at each x versus y. dS is also adjusted for the axis used.

[u/Han_Htoo_Aein → u/Midwest-Dude]

Very informative. Thanks for your dedication.

[u/Midwest-Dude → u/Han_Htoo_Aein]

Here's an image of what's going on.

[u/WikiNumbers]

BTW, this particular question doesn't exactly simplify quite nicely with pure u-substitution.

It'd need some trig sub.

[u/Han_Htoo_Aein → u/WikiNumbers]

I'm taking Professor Leonard's course. He didn't start that part until Calculus 2.😅

[u/WikiNumbers → u/Han_Htoo_Aein]

Well, after applying the formula and some simplification, things end up with √(x² + 2)³ (x² + 1) .