Chapter 10 - Parametric Equations and Polar Coordinates - 10.2 Calculus with Parametric Curves - 10.2 Exercises - Page 696: 59

$S\approx10.6705$

$x$ = $t+e^t$ $y$ = $e^{-t}$ $\frac{dx}{dt}$ = $1+e^t$ $\frac{dy}{dt}$ = $-e^{-t}$ $\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2 = (1+e^t)^2+(-e^{-t})^2 = 1+2e^t+e^2t+e^{-2t}$ $S$ = $\int2{\pi}yds$ $S$ = $\int_0^1{2{\pi}e^{-t}\sqrt {1+2e^t+e^2t+e^{-2t}}}dt$ $S$ $\approx$ $10.6705$