How do you find the length of the polar curve $r = θ$ $r=theta$ ?

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How do you find the length of the polar curve $r = θ$ $r=theta$ ?

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Answer 1 · 2014-09-07T15:02:26

If $θ$ $theta$ goes from $θ_{1}$ $theta_1$ to $θ_{2}$ $theta_2$ , then the arc length of can be found by
$L = \int_{θ_{1}}^{θ_{2}} \sqrt{θ^{2} + 1} d θ$ $L=int_{theta_1}^{theta_2}sqrt{theta^2+1}d theta$ .

Since $r = θ$ $r=theta$ , which gives $\frac{d r}{d θ} = 1$ ${dr}/{d theta}=1$ , we have
$L = \int_{θ_{1}}^{θ_{2}} \sqrt{r^{2} + {(\frac{d r}{d θ})}^{2}} d θ = \int_{θ_{1}}^{θ_{2}} \sqrt{θ^{2} + 1} d θ$ $L=int_{theta_1}^{theta_2}sqrt{r^2+({dr}/{d theta})^2}d theta =int_{theta_1}^{theta_2}sqrt{theta^2+1}d theta$

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How do you find the length of the polar curve $r = θ$ $r=theta$ ?

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