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Find the exact length of the curve.

$ y = \ln (\cos x) $ , $ 0 \le x \le \frac{\pi}{3} $

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$\ln (2+\sqrt{3})$

03:19

Harriet O'Brien

Calculus 2 / BC

Chapter 8

Further Applications of Integration

Section 1

Arc Length

Applications of Integration

Harvey Mudd College

University of Nottingham

Idaho State University

Lectures

02:29

Find the exact length of t…

01:42

Arc length Find the length…

03:38

02:34

03:04

09:39

Find the are length of the…

Okay, so our goal is to find the arc length of this function. L. N. Co sign of acts. We are interested in finding it from X. Goes to zero to X. Goes to pi over three. Um So let's review general equation for arc length. So our length is the integral from A to B. of the square root of one plus the derivative A squared D. X. So let's go ahead and find the derivative. So if Y is L. N. Cosin of acts, then the derivative will be by chain will first the derivative of Ellen which gives me one over cosine effects. But then I multiply by the derivative of cosine effects and I get times minus sign up. So we see that Y prime is equal to minus tan of X. So let's go ahead and substitute that in. And we also have our limits. We're going to integrate from zero two pi over three square root of one plus minus tan of X quantity squared. The okay, so the nice thing is I can clean this up a little bit and inside the radical I get one plus tan of X squared but that is a known um trig identity. That is equal to seek it square. So therefore I get square, read a second squared and I get just um the integral of second of FCS. Now this one is not easy to solve. Um it's not intuitive so usually you would use a trick table. Sorry uh integral table and the anti drift turns out to be Ln of seeking effects plus tan effects. Um Alright so we plug in our limits And that will give us LNA seek it high over three plus a tan A pi over three helped us attract and we gotta plug in R. Zero so seek it. Zero Plus 10 of zero. Let's see what we get for our final answer. Almost there. So let's see seeking a pi over three Co sign would be a half switch, flip it for two. Tana Pi over three is signed over co sign. So that would be route three Over to over half. So we just get route three. Let's see seeking to zero is one. Tana, zero is zero but Ln of one is zero. So our final answer. Drum roll is Ln of two plus square root of three. And we solved it So congratulations. Have a great day and see you next time.

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