Answer
$1.552250$
Work Step by Step
With $y=\sqrt{1-x^{2}},$ the five points along the curve are
$$
P_{0}(0,1), P_{1}(1 / 4, \sqrt{15} / 4), P_{2}(1 / 2, \sqrt{3} / 2), P_{3}(3 / 4, \sqrt{7} / 4), P_{4}(1,0)
$$
Then
\begin{aligned}
&\overline{P_{0} P_{1}}=\sqrt{\frac{1}{16}+\left(\frac{4-\sqrt{15}}{4}\right)^{2}} \approx 0.252009\\
&\overline{P_{1} P_{2}}=\sqrt{\frac{1}{16}+\left(\frac{2 \sqrt{3}-\sqrt{15}}{4}\right)^{2}} \approx 0.270091\\
&\overline{P_{2} P_{3}}=\sqrt{\frac{1}{16}+\left(\frac{2 \sqrt{3}-\sqrt{7}}{4}\right)^{2}} \approx 0.323042\\
&\overline{P_{3} P_{4}}=\sqrt{\frac{1}{16}+\frac{7}{16}} \quad \approx 0.707108
\end{aligned}
and the total approximate distance is $1.552250$, whereas $\pi/2\approx 1.57079$.
