Derive the formula for the surface area of a spherical

Derive the formula for the surface area of a spherical cap of height $h$ on a sphere of radius $R$.

Place the cap so it lies over $x\in[R-h,R]$ on the semicircle $y=\sqrt{R^2-x^2}$.

Step 1: Reuse the simplified integrand

As in Problem 118, $y\sqrt{1+(y')^2}=R$ (a constant), so

\[S=2\pi\int_{R-h}^{R}R\,dx = 2\pi R\big[R-(R-h)\big] = 2\pi Rh\]

\[\boxed{S=2\pi Rh}\]

Step 2: Verify special cases

Hemisphere ($h=R$): $S=2\pi R^2$, exactly half the full sphere's area. ✓
Full sphere ($h=2R$): $S=4\pi R^2$, matching Problem 118. ✓

Step 3: Cross-check via spherical coordinates

The cap corresponds to $\phi\in[0,\phi_0]$ with $\cos\phi_0=1-\frac hR$:

\[S=2\pi R^2\int_0^{\phi_0}\sin\phi\,d\phi = 2\pi R^2(1-\cos\phi_0) = 2\pi R^2\cdot\frac hR = 2\pi Rh\quad\checkmark\]

Step 4: Interpretation

The formula $S=2\pi Rh$ depends only on the cap's height, not its position on the sphere — it equals the lateral area of a cylinder of radius $R$ and height $h$, another instance of Archimedes' cylinder principle.