Student Question

`y=ln(sinx) , [pi/4 , (3pi)/4]` Find the arc length of the graph of the function over the indicated interval.

Expert Answers

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The arc length of a function of x, f(x), over an interval is determined by the formula below:


So using the function given, let us first find `(dy)/(dx):`


We can now substitute this into our formula above:


Which can then be simplified to:


Then you find the definite integral as you normally would.  (Using the method shown on the link below, you can find the integral of csc(x).)




Here, we will switch the two natural logarithm terms and use the quotient property to combine them into a single log:


If you rationalize the denominator (by multiplying by the conjugate and simplifying) and use the power property of logs, you are left with:


So the exact value of the arc length of the graph of the function over the given interval is `2ln|sqrt(2)+1|`

which is approximately equal to 1.76.

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