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Chapter 8: Problem 90

Arc Length Find the arc length of the graph of \(y=\ln (\cos x)\) from \(x=0\) to \(x=\pi / 3\)

Short Answer

Expert verified
The arc length of the graph of \(y = \ln( \cos x )\) from \(x = 0\) to \(x = \pi / 3\) is \(\ln 2\)

Step by step solution

01

Find the Derivative of the Function

The derivative of \(y = \ln( \cos x )\) can be calculated using the chain rule: \(f'(x) = \frac{-\sin x}{\cos x}=-\tan x\)
02

Square the Derivative and add 1

Next, square the derivative and add 1: \(1 + [f'(x)]^2 = 1 + \tan^2x = \sec^2x \), using the identity \(1 + \tan^2x = \sec^2x \)
03

Find the Square Root of the Result

The square root of \(\sec^2x\) is \(|\sec x|\). Since \(x\) is in the range of 0 to \(\pi / 3\), both \(\cos x\) and its reciprocal \(\sec x\) are positive. So, the square root of \(\sec^2x\) is just \(\sec x\)
04

Integration to Find the Arc Length

The arc length is given by the formula \(L = \int_{a}^{b}\sqrt{1+ [f'(x)]^2} dx = \int_{0}^{\pi / 3} \sec x dx \). This leads to \(L = \left[ \ln |\sec x + \tan x| \right]_0^{\pi / 3} = \ln 2 \)

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