Calculus Calculus Math Forum

 February 12th, 2017, 03:14 PM #1 Senior Member   Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 Question about Arch Length Find the arc length of r(t) = < t^2, sin(t), cos(t) >, 0 <= t <= 1 Answer: sqrt(5) / 2 + ln( sqrt(5) + 2 ) / 4 I use the standard fomula: Integral( |r'(t)|) between 0 and 1, but I keep getting the following incorrect result: ( 2 * sqrt(5) - ln ( sqrt(5) - 2)) / 4 I would be much appreciated if you could point me at the right direction... February 12th, 2017, 04:39 PM #2 Math Team   Joined: Jul 2011 From: Texas Posts: 2,949 Thanks: 1555 $\displaystyle \int_0^1 \sqrt{4t^2+1} \, dt$ $2t = \tan{\theta}$ $2 \, dt = \sec^2{\theta} \, d\theta$ $\displaystyle \dfrac{1}{2}\int_0^{\phi} \sec^3{\theta} \, d\theta$, where $\phi = \arctan(2)$ using integration by parts, note ... $\displaystyle \int \sec^3{\theta} \, d\theta = \sec{\theta}\tan{\theta} - \int \sec{\theta}\tan^2{\theta} \, d\theta$ $\displaystyle \int \sec^3{\theta} \, d\theta = \sec{\theta}\tan{\theta} - \int \sec{\theta}(\sec^2{\theta}-1) \, d\theta$ $\displaystyle \int \sec^3{\theta} \, d\theta = \sec{\theta}\tan{\theta} - \int \sec^3{\theta}-\sec{\theta} \, d\theta$ $\displaystyle \int \sec^3{\theta} \, d\theta = \sec{\theta}\tan{\theta} - \int \sec^3{\theta}\, d\theta + \int \sec{\theta} \, d\theta$ $\displaystyle 2\int \sec^3{\theta} \, d\theta = \sec{\theta}\tan{\theta} + \int \sec{\theta} \, d\theta$ $\displaystyle \int \sec^3{\theta} \, d\theta = \dfrac{1}{2}\bigg[\sec{\theta}\tan{\theta} + \int \sec{\theta} \, d\theta\bigg]$ therefore ... $\displaystyle \dfrac{1}{2}\int_0^{\phi} \sec^3{\theta} \, d\theta = \dfrac{1}{4}\bigg[\sec{\theta}\tan{\theta} + \ln|\sec{\theta}+\tan{\theta}|\bigg]_0^{\phi}$ $\displaystyle \dfrac{1}{2}\int_0^{\phi} \sec^3{\theta} \, d\theta = \dfrac{1}{4}\bigg[2\sqrt{5} + \ln(\sqrt{5}+2)\bigg]$ Thanks from topsquark February 13th, 2017, 03:27 PM #3 Senior Member   Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 May I ask what gives you the clue for u-substitution 2t = tan(x) February 13th, 2017, 05:26 PM   #4
Math Team

Joined: Jul 2011
From: Texas

Posts: 2,949
Thanks: 1555

Quote:
 Originally Posted by zollen May I ask what gives you the clue for u-substitution 2t = tan(x)
note $4x^2 + 1 =(2x)^2 + 1$

compare $(2x)^2+1$ to $\tan^2{x} + 1$ ... Tags arch, length, question Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post fredlo2010 Calculus 6 November 4th, 2014 03:41 AM fredlo2010 Calculus 0 November 2nd, 2014 05:12 PM gen_shao Physics 1 August 11th, 2014 05:09 AM Trav44 Algebra 5 October 29th, 2013 04:01 AM hatcher777 Algebra 4 February 17th, 2007 02:29 AM

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