On a \(pV\) diagram, the area under the curve represents the work \(W\) done by the gas; Figure 15-7b shows this area for an isothermal expansion. Unlike the isobaric case shown in Figure 15-6, the area under the curve is not a rectangle, so we cannot find \(W\) simply by multiplying the pressure and the change in volume. Using calculus, it can be shown that the work that the gas does as the volume changes is