Area of Surface of Revolution/Parametric Form

Theorem

Let $\SS$ be a surface of revolution such that:

$\SS$ is embedded in a Cartesian $3$-space

the axis of revolution of $\SS$ is aligned with the $x$-axis

the curve $\CC$ being rotated to generate $\SS$ is the plane curve described by the parametric equations:

$\ds \quad \ \ $

$\ds x$

$=$

$\ds \map x t$

$\ds y$

$=$

$\ds \map y t$

where $t$ is in the closed interval $\closedint a b$.

Then the area of $\SS$ is given by:

$\ds \map \AA \SS = 2 \pi \int_a^b y \sqrt {\paren {\dfrac {\d x} {\d t} }^2 + \paren {\dfrac {\d y} {\d t} }^2} \rd t$

Proof

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Sources

2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): area of a surface of revolution
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): area of a surface of revolution