= Surface area of revolution of a centrally symmetric curve
{title2=$A=2\pi dL$}
Let a simple closed regular planar <curve> be invariant under <point reflection> through a center and remain strictly on one side of a coplanar <axis of rotation>, at <distance> $d$ from its center. If $L$ is its <arc length>, its <surface of revolution> has <surface area> $2\pi dL$. To prove this, choose coordinates with the center at the origin and the <axis of rotation> at $y=-d$. The <surface area> formula is $2\pi\oint(d+y)\,ds$. Pair each point with its reflected point: the <arc length> element $ds$ is unchanged and $y$ changes sign. Thus $\oint y\,ds=0$, proving the result. The one-side condition prevents the <surface> from passing through the <axis of rotation> or requiring an absolute value with changing sign.
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