Answer
$L=\int_0^{\frac{\pi}{2}}\sqrt{1+(1-\sin(x))^2}dx=1.7294$
Work Step by Step
More than 1.67, line length $\sqrt{\Big(\frac{\pi}{2}\Big)^2+\Big(\frac{\pi}{2}+\cos\Big(\frac{\pi}{2}\Big)-1\Big)^2}=1.67$
$L=\int_a^b\sqrt{1+\Big(\frac{dy}{dx}\Big)^2}dx$
$y=x+\cos(x)$
$\frac{dy}{dx}=1-\sin(x)$
$a=0$, $b={\frac{\pi}{2}}$
$L=\int_0^{\frac{\pi}{2}}\sqrt{1+(1-\sin(x))^2}dx=1.7294$
