Example 2.3.4: Area of a rectangle

The rectangle is the set $\{x=0,0\le y\le a,0\le z\le b\}$. On a surface $x=\text{const}$ the area element is $dS=dydz$, from Eq. (2.3.11). The limits are constants, so the double integral factorises:

This is of course the familiar formula. The point of the example is the method: choose the correct area element for the surface, then let the limits describe its extent.