${\displaystyle dx}$

The radius of the circular ring is${\displaystyle f(x)=\sqrt{r^2-x^2}}$. The length of the circular ring is equal to${\displaystyle 2\pi \cdot f(x)}$
The width of the ring can be determined by usingPythagoras' formula for a rectangular triangle with side lengths${\displaystyle dx}$ and${\displaystyle f^{\prime }(x)\cdot dx}$, which leads to${\displaystyle \sqrt{1+f^{\prime }(x{)}^2}dx}$
The infinitesimal surface area of the circular ring thus is equal to${\displaystyle 2\pi f(x)\cdot \sqrt{1+f^{\prime }(x{)}^2}dx}$
Thederivative of${\displaystyle f(x)}$ is equal to${\displaystyle f^{\prime }(x)=\frac{-x}{\sqrt{r^2-x^2}}}$
The surface area of the sphere can be calculated as

${\displaystyle {\int }_{-r}^{r}2\pi f(x)\cdot \sqrt{1+f^{\prime }(x{)}^2}dx}$ =${\displaystyle {\int }_{-r}^{r}2\pi \sqrt{r^2-x^2}\cdot \sqrt{(}1+\frac{x^2}{r^2-x^2})dx={\int }_{-r}^{r}2\pi \sqrt{r^2}dx=2\pi r{\int }_{-r}^{r}1dx}$

Theantiderivative needed is the simple linear function${\displaystyle x}$
Thus, the sphere surface area amounts to

A_sphere =${\displaystyle 2\pi r[r-(-r)]=4\pi r^2}$

http://web.archive.org/web/20120427201949/http://www.math.usma.edu/people/rickey/hm/CalcNotes/schwarz-paradox.pdf— Precedingunsigned comment added by94.197.120.122 (talk)22:10, 9 October 2013 (UTC)[reply]

http://web.archive.org/web/20111215152255/http://mathdl.maa.org/images/upload_library/22/Polya/00494925.di020678.02p0385w.pdf— Precedingunsigned comment added by94.197.120.122 (talk)22:19, 9 October 2013 (UTC)[reply]

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The technical term "surface measure" doesn't have a wikipedia page. Is this the page? Should a redirect come here?

The article mentions a characterization of surface area using a few properties such as additivity and invariance under Euclidean motions. There should be given a source for this, and preferably it should also be explained more clearly.MathHisSci (talk)21:52, 14 April 2023 (UTC)[reply]