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:

`L=int_a^bsqrt(1+((dy)/(dx))^2)dx`

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

`d/(dx)(ln(sin(x)))=(1/(sin(x)))*(cos(x))=(cos(x))/(sin(x))=cot(x)`

We can now substitute this into our formula above:

`L=int_a^bsqrt(1+((dy)/(dx))^2)dx=int_(pi/4)^((3pi)/4)sqrt(1+(cot(x))^2)dx`

Which can then be simplified to:

`L=int_(pi/4)^((3pi)/4)sqrt(1+cot^2(x))dx=int_(pi/4)^((3pi)/4)sqrt(csc^2(x))dx=int_(pi/4)^((3pi)/4)csc(x)dx`

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).)

`L=int_(pi/4)^((3pi)/4)csc(x)dx=-ln|csc(x)+cot(x)|_(pi/4)^((3pi)/4)`

`L=-ln|csc((3pi)/4)+cot((3pi)/4)|-(-ln|csc(pi/4)+cot(pi/4)|)`

`L=-ln|sqrt(2)+(-1)|-(-ln(sqrt(2)+1|)=-ln|sqrt(2)-1|+ln|sqrt(2)+1|`

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

`L=ln|sqrt(2)+1|-ln|sqrt(2)-1|=ln|(sqrt(2)+1)/(sqrt(2)-1)|`

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

`L=ln|(sqrt(2)+1)^2/1|=ln|(sqrt(2)+1)^2|=2ln|sqrt(2)+1|`

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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