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When gas expands in a cylinder with radius $ r $, the pressure at any given time is a function of the volume: $ P = P(V) $. The force exerted by the gas on the piston (see the figure) is the product of the pressure and the area: $ F = \pi r^2 P $. Show that the work done by the gas when the volume expands from volume $ V_1 $ to volume $ V_2 $ is

$$ W = \int_{V_1}^{V_2} P dV $$

$\int_{V_{1}}^{V_{2}} P(V) d V$

Applications of Integration

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Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

so prove that force equals two anywhere from we want to read to you the so first we're gonna start off by defined the volume which is close to the area times Glenn's. So why are square eggs and, uh but the definition of horse, which is x two x want to x toole after backs the X But after what happened, this here is actually Peter is being so the pressure on the bottom, right? Says pious, quite sounds pee wee ettes Tom's dia's piece of blue for the pressure We know that, uh, re X equals to the by r squared. It's so the V that's equals two by r squared the ends and they used this relation to substitute this this integral. We're gonna under the voice, be one to be too p of be TV. This far is simply be me. And that's proved over Proposition

University of Illinois at Urbana-Champaign

Applications of Integration