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Chapter 8: Problem 89

Arc Length Find the arc length of the graph of \(y=\ln (\sin x)\) from \(x=\pi / 4\) to \(x=\pi / 2\)

Short Answer

Expert verified
The exact solution cannot be expressed with basic mathematical functions or constants. The solution is a numeric value approximately equal to 0.443

Step by step solution

01

Find the derivative of the function

First, find the derivative of \(y = \ln(\sin(x))\). Using the chain rule for derivatives, where \((f(g(x)))' = f'(g(x)) \cdot g'(x)\), the derivative of \(\sin(x)\) is \(\cos(x)\), and the derivative of \(\ln(x)\) is \(1/x\), we arrive at: \(y' = \frac{1}{\sin(x)} \cdot \cos(x) = \cot(x)\).
02

Insert the derivative into the formula for arc length

Next, insert the derivative \(\cot(x)\) into the formula for arc length: \[L = \int_{\pi / 4}^{\pi / 2} \sqrt{1 + \cot^2(x)} \, dx\] After simplifying \(\sqrt{1 + \cot^2(x)}\) to \(\csc(x)\) (this is equivalent due to the Pythagorean identity \(\cot^2(x) + 1 = \csc^2(x)\)), we get: \[L = \int_{\pi / 4}^{\pi / 2} \csc(x) \, dx\]
03

Evaluate the integral

Now, evaluate the above integral. The antiderivative of \(-\csc(x)\) is \(\ln|\csc(x) + \cot(x)|\). Remember, when taking the antiderivative, the range is from \(\pi / 4\) to \(\pi / 2\), so substitute these values in: \[L = \ln|\csc(\pi / 2) + \cot(\pi / 2)| - \ln|\csc(\pi / 4) + \cot(\pi / 4)|\] Simplify by inserting the values for \(\csc\) and \(\cot\) and calculate the result.

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