Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 9 - Further Applications of the Integral and Taylor Polynomials - 9.1 Arc Length and Surface Area - Exercises - Page 469: 33

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 $\pi/2\approx 1.57079$.
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