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Consider the plane curve $\gamma$ in polar coordinates: $$ r=r_0+e^{\lambda\theta}, \quad \theta_1 \le \theta \le \theta_2, $$ where $r_0,\lambda,\theta_1>0$. Is it possible to compute explicitly the length of $\gamma$?

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    Use $\int_a^b\|\gamma'(t)\|dt$, where $\gamma(t)=(x(t),y(t))$, with the usual change of coordinates.2012-07-17
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    @Sigur I know that formula, it doesn't answer my question!2012-07-17
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    Why not? What did you think?2012-07-17
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    @Sigur The question seems clear enough! I'm not saying that I need the formula for the length a curve.2012-07-17
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    @Mercy: How does the formula not answer your question? Plug in your parametric equation and solve the integral -- symbolically if you can, numerically if you must. That's how you compute it. (In other words, the _explicit_ answer to your question is "yes").2012-07-17
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    @HenningMakholm Read carefully please. I'm asking whether there is a closed form for "an integral"!2012-07-17
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    @Mercy: Your question as currently written does not mention anything about closed forms.2012-07-17
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    @Mercy Actually, your question is "Is it possible to compute explicitly the length of $\gamma$?" The answer is "yes" and some posters have given some clues as how to go about the calculation.2012-07-17

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