Answer
$L$ $\approx$ $612.3053$
Work Step by Step
$x$ = $t-e^t$
$y$ = $t+e^t$
$\frac{dx}{dt}$ = $1-e^t$
$\frac{dy}{dt}$ = $1+e^t$
$\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2$ = $(1-e^t)^2+(1+e^t)^2$ = $2+2e^{2t}$
$L$ = $\int_{-6}^{6}\sqrt {2+2e^{2t}}dt$
$f(t)$ = $\sqrt {2+2e^{2t}}$
Simpson's rule with $n$ = $6$
$Δt$ = $\frac{6-(-6)}{6}$ = $2$
$L$ = $\frac{2}{3}[f(-6)+4f(-4)+2f(-2)+4f(0)+2f(2)+4f(4)+f(6)]$
$L$ $\approx$ $612.3053$
