Derive the formula for the surface area of a sphere

Derive the formula for the surface area of a sphere of radius $R$.

We rotate a semicircle about a diameter to generate the sphere.

Step 1: Set up

$y=\sqrt{R^2-x^2}$ for $x\in[-R,R]$, rotated about the $x$-axis.

Step 2: Differentiate

\[y'=-\frac x{\sqrt{R^2-x^2}} \implies 1+(y')^2=\frac{R^2-x^2+x^2}{R^2-x^2}=\frac{R^2}{R^2-x^2}\]

\[\sqrt{1+(y')^2}=\frac R{\sqrt{R^2-x^2}}\]

Step 3: Set up the surface integral

\[S=2\pi\int_{-R}^{R}\sqrt{R^2-x^2}\cdot\frac R{\sqrt{R^2-x^2}}\,dx = 2\pi R\int_{-R}^{R}dx = 2\pi R\cdot2R\]

\[\boxed{S=4\pi R^2}\]

Step 4: Cross-check via spherical coordinates

With surface element $dS=R^2\sin\phi\,d\phi\,d\theta$:

\[S=\int_0^{2\pi}\int_0^\pi R^2\sin\phi\,d\phi\,d\theta = 2\pi R^2\big[-\cos\phi\big]_0^\pi = 2\pi R^2\cdot2 = 4\pi R^2\quad\checkmark\]

Step 5: Archimedes' cylinder

The circumscribed cylinder (radius $R$, height $2R$) has lateral area $2\pi R\cdot2R=4\pi R^2$ — exactly the sphere's surface area, a fact Archimedes considered among his finest discoveries.