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The integral is used for deriving the equation for finding work in a reversible gas expansion. The integral is used to find the area of the curve in a volume vs. pressure graph (you can find that on p 266 of the textbook). The graph is curve because in a reversible gas expansion, the external pressure is reduced infinitesimally, and the area underneath the curve is the work since PV= w (and it's a volume vs pressure graph). Hope that made sense!
We used integrals to derive the work equations because when systems are at equilibrium, changes within them occur infinitesimally. The integral allows us to sum together all of these small steps to calculate the total work. As mentioned above, the integrals were only used for the formula derivations and are not needed for actual work calculations.
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