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Thread: Arc Length - Polar Coordinates

  1. #1
    Del
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    Arc Length - Polar Coordinates

    Find the length of the spiraling polar curve :

    r = 5 e^2θ

    From 0 to 2pi

    Please help!
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  2. #2
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    Hello, Del!

    Find the length of the spiraling polar curve: .$\displaystyle r \:= \:5e^{2\theta}$ .from 0 to $\displaystyle 2\pi$
    Formula: . $\displaystyle L \;=\;\int^{\beta}_{\alpha}\sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\,d\theta$


    We have: . $\displaystyle \frac{dr}{d\theta} \:=\:10e^{2\theta}$

    Then: . $\displaystyle r^2 + \left(\frac{dr}{d\theta}\right)^2 \;=\;\left(5e^{2\theta}\right)^2 + \left(10e^{2\theta}\right)^2 \;=\;25e^{4\theta} + 100e^{4\theta} \;=\;125e^{4\theta}$

    Hence: . $\displaystyle \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \;=\;5\sqrt{5}\,e^{2\theta} $


    Therefore: . $\displaystyle L \;=\;5\sqrt{5}\int^{2\pi}_0 e^{2\theta}\,d\theta$

    Got it?

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