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I believe Lavelle will simplify the integral equation next lecture. In the textbook, there is an equation that is much easier to work with--I believe he will be discussing that. This one was used when discussing reversible, isothermal expansion reactions: the ones done in small steps (hence the integral). P is constant but volume is changing. Expansion of gas held at constant pressure is represented by the equation w=p times delta v.
I think you should just know that an integral is used to sum a series of steps, and in this case infinitesimal changes. It's basically the area under a curve, so the reason we use it to sum up infinitesimal changes is so we can sum the change in a pressure/volume curve.
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