Page:Calculus Made Easy.pdf/235

(3) Here’s another case–that of the die-away curve (p. 156). Find the area between $x=0$ and $x=a$, of the curve (Fig. 57) whose equation is

${\begin{aligned}y&=b\epsilon ^{-x}.\\{\text{Area}}&=b\int _{x=0}^{x=a}\epsilon ^{-x}\times dx.\\\end{aligned}}$

The integration (p. 201) gives

${\begin{aligned}&=b\left[-\epsilon ^{-x}\right]_{0}^{a}\\&=b{\bigl [}-\epsilon ^{-a}-(-\epsilon ^{-0}){\bigr ]}\\&=b(1-\epsilon ^{-a}).\end{aligned}}$

Fig. 57.	Fig. 58.

(4) Another example is afforded by the adiabatic curve of a perfect gas, the equation to which is $pv^{n}=c$, where $p$ stands for pressure, $v$ for volume, and $n$ is of the value $1.42$ (Fig. 58).

Find the area under the curve (which is proportional