Example 2.3.13: Lateral area of a cylinder

The lateral (side) surface is $\{\rho =b,0\le \varphi <2\pi ,0\le z\le h\}$, a piece of the coordinate surface $\rho =\text{const}$. Its area element is $dS=\rho d\varphi dz$ evaluated at $\rho =b$:

This is the circumference of Example 2.3.11 times the height, as expected if the curved surface is cut along a vertical line and unrolled into a $2\pi b\times h$ rectangle.