Area of Surface of Revolution

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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 $y = \map f x$
$\CC$ has endpoints at $x = a$ and $x = b$.


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

$\map \AA \SS = \ds \int_a^b 2 \pi y \sqrt {1 + \paren {\dfrac {\d y} {\d x} }^2} \rd x$


Proof


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Sources

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