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Thread: Find the length of the curve

  1. #1
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    Find the length of the curve

    Find the length of the curve $\displaystyle r=\frac{e^\theta}{\sqrt2}$ on the interval $\displaystyle 0\leq\theta\leq\pi$. I've plugged everything into the equation, but I can't seem to get the same answer that is on the answer sheet, which is $\displaystyle e^\pi-1$.
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    Re: Find the length of the curve

    the answer is $\displaystyle e^{\pi-1}$ not $\displaystyle (e^\pi - 1)$ as you've written.

    does that help any?
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    Re: Find the length of the curve

    No, could you go over it step-by-step?
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    Re: Find the length of the curve

    Do you have a rule that you can follow to find the arclength of a polar curve?
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    Re: Find the length of the curve

    Yes, the equation is $\displaystyle L=\int_{a}^{b}\sqrt{r^2+(\frac{dr}{d\theta})^2$.
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    Re: Find the length of the curve

    Well then, what is your $\displaystyle \displaystyle \begin{align*} r^2 \end{align*}$ and $\displaystyle \displaystyle \begin{align*} \left( \frac{dr}{d\theta} \right) ^2 \end{align*}$? What are your a and b?

    Also, the equation is actually $\displaystyle \displaystyle \begin{align*} L = \int_a^b{\sqrt{r^2 + \left( \frac{dr}{d\theta} \right) ^2 }\,d\theta} \end{align*}$...
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    Re: Find the length of the curve

    $\displaystyle r^2=\frac{dr}{d\theta}=\frac{e^{\theta}}{\sqrt{2}}$;$\displaystyle a=0$;$\displaystyle b=\pi$
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    Re: Find the length of the curve

    Quote Originally Posted by accountholder View Post
    $\displaystyle r^2=\frac{dr}{d\theta}=\frac{e^{\theta}}{\sqrt{2}}$;$\displaystyle a=0$;$\displaystyle b=\pi$
    no.

    you are given $\displaystyle r=\frac{e^\theta}{\sqrt{2}}$

    so $\displaystyle r^2=\frac{e^{2\theta}}{2}$

    $\displaystyle \frac{d^2r}{d\theta^2}=\frac{dr}{d\theta}=\frac{e^ \theta}{\sqrt{2}}$

    so $\displaystyle \left(\frac{d^2r}{d\theta^2}\right)^2=\frac{e^{2 \theta}}{2}$

    now take your arc length formula and do the integration. Your limits are correct.
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