Chapter 18: Problem 62

Experimental studies show that the \(p V\) curve for a frog's lung can be approximated by \(p=10 v^{3}-67 v^{2}+220 v,\) with \(v\) in \(\mathrm{mL}\) and \(p\) in \(\mathrm{Pa}\). Find the work done when such a lung inflates from zero to \(4.5 \mathrm{mL}\) volume.

Short Answer

Expert verified

To have the work done, firstly, the integral expression for work has to be written, then the integral must be evaluated and finally, the value of the expression must be calculated by substituting the values.

Step by step solution

Writing the integral expression for work

As work done is given by the integral of pressure with respect to volume,\(W=\int_{V_{1}}^{V_{2}} p d V\), this integral turns into \( W=\int_{0}^{4.5} (10 v^{3}-67 v^{2}+220 v) dv\).

Evaluate the integral

We now evaluate this integral using the fundamental theorem of calculus. We first find the antiderivative (F) of \(10 v^{3}-67 v^{2}+220 v\), which is \(\frac {10} {4} v^{4}-\frac {67} {3} v^{3}+110 v^{2}\). So, \( W=\left[\frac {10} {4} v^{4}-\frac {67} {3} v^{3}+110 v^{2}\right]_{0}^{4.5}\).

Calculate the value

Insert the values into the definite integral: \( W=\frac {10} {4}*(4.5)^{4}-\frac {67} {3} *(4.5)^3+110 *(4.5)^2 - (\frac {10} {4}*0^{4}-\frac {67} {3}*0^{3}+110*0^{2})\). Upon evaluating this expression, we get the work done.

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Experimental studies show that the \(p V\) curve for a frog's lung can be approximated by \(p=10 v^{3}-67 v^{2}+220 v,\) with \(v\) in \(\mathrm{mL}\) and \(p\) in \(\mathrm{Pa}\). Find the work done when such a lung inflates from zero to \(4.5 \mathrm{mL}\) volume.

Short Answer

Step by step solution

Writing the integral expression for work

Evaluate the integral

Calculate the value

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