Question

### Gauthmathier7782

Grade 10 · 2021-01-18

Find the length to the polar curve r=\sin \theta -\cos \theta , when 0\leq \theta \leq \pi .（ ）

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Answer

4.8(126) votes

### Gauthmathier7954

Grade 10 · 2021-01-18

Answer

B

Explanation

Observe that r^{2}=(\sin \theta -\cos \theta )^{2}=\sin ^{2}\theta -2\sin \theta \cos \theta +\cos ^{2}\theta =1-\sin (2\theta ), and

\left(\dfrac {\d r}{\d\theta }\right)^{2}=(\cos \theta +\sin \theta )^{2}=1+\sin (2\theta ), Therefore, the arc length is given by

\int\limits _{0}^{\pi }\sqrt {r^{2}+\left(\dfrac {\d r}{\d\theta }\right)^{2}}\d\theta =\int\limits _{0}^{\pi }\sqrt {2}=\sqrt {2}\pi

\left(\dfrac {\d r}{\d\theta }\right)^{2}=(\cos \theta +\sin \theta )^{2}=1+\sin (2\theta ), Therefore, the arc length is given by

\int\limits _{0}^{\pi }\sqrt {r^{2}+\left(\dfrac {\d r}{\d\theta }\right)^{2}}\d\theta =\int\limits _{0}^{\pi }\sqrt {2}=\sqrt {2}\pi

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