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 $\gamma$?
Length of a plane curve in polar coordinate
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integration
definite-integrals
plane-curves
polar-coordinates
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0Use $\int_a^b\|\gamma'(t)\|dt$, where $\gamma(t)=(x(t),y(t))$, with the usual change of coordinates. – 2012-07-17
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0@Sigur I know that formula, it doesn't answer my question! – 2012-07-17
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0Why not? What did you think? – 2012-07-17
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0@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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0@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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0@HenningMakholm Read carefully please. I'm asking whether there is a closed form for "an integral"! – 2012-07-17
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0@Mercy: Your question as currently written does not mention anything about closed forms. – 2012-07-17
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1@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