Mr. Integral
Math notes/Calculus II/Geometry and Physical Applications

Arc Length and Surface Area — Cheat sheet

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Core rule

ds=1+(y′)2 dx,L=∫ds,S=∫2πr ds.ds=\sqrt{1+(y')^2}\,dx,\quad L=\int ds,\quad S=\int2\pi r\,ds.

Watch for

Surface area is not volume and excludes end caps unless added separately. Radius is a nonnegative distance.