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

Friday, February 4, 2011

$\displaystyle{y}={\ln{{\left({\sin{{x}}}\right)}}},{\left[\frac{\pi}{{4}},\frac{{{3}\pi}}{{4}}\right]}$ Find the arc length of the graph of the function over the indicated interval.

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

$\displaystyle{L}={\int_{{a}}^{{b}}}\sqrt{{{1}+{{\left(\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}\right)}}^{{2}}}}{\left.{d}{x}\right.}$

So using the function given, let us first find $\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}:$

$\displaystyle\frac{{d}}{{{\left.{d}{x}\right.}}}{\left({\ln{{\left({\sin{{\left({x}\right)}}}\right)}}}\right)}={\left(\frac{{1}}{{{\sin{{\left({x}\right)}}}}}\right)}\cdot{\left({\cos{{\left({x}\right)}}}\right)}=\frac{{{\cos{{\left({x}\right)}}}}}{{{\sin{{\left({x}\right)}}}}}={\cot{{\left({x}\right)}}}$

We can now substitute this into our formula above:

$\displaystyle{L}={\int_{{a}}^{{b}}}\sqrt{{{1}+{{\left(\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}\right)}}^{{2}}}}{\left.{d}{x}\right.}={\int_{{\frac{\pi}{{4}}}}^{{\frac{{{3}\pi}}{{4}}}}}\sqrt{{{1}+{{\left({\cot{{\left({x}\right)}}}\right)}}^{{2}}}}{\left.{d}{x}\right.}$

Which can then be simplified to:

$\displaystyle{L}={\int_{{\frac{\pi}{{4}}}}^{{\frac{{{3}\pi}}{{4}}}}}\sqrt{{{1}+{{\cot}}^{{2}}{\left({x}\right)}}}{\left.{d}{x}\right.}={\int_{{\frac{\pi}{{4}}}}^{{\frac{{{3}\pi}}{{4}}}}}\sqrt{{{{\csc}}^{{2}}{\left({x}\right)}}}{\left.{d}{x}\right.}={\int_{{\frac{\pi}{{4}}}}^{{\frac{{{3}\pi}}{{4}}}}}{\csc{{\left({x}\right)}}}{\left.{d}{x}\right.}$

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).)

$\displaystyle{L}={\int_{{\frac{\pi}{{4}}}}^{{\frac{{{3}\pi}}{{4}}}}}{\csc{{\left({x}\right)}}}{\left.{d}{x}\right.}=-{\ln}{{\left|{\csc{{\left({x}\right)}}}+{\cot{{\left({x}\right)}}}\right|}_{{\frac{\pi}{{4}}}}^{{\frac{{{3}\pi}}{{4}}}}}$

$\displaystyle{L}=-{\ln}{\left|{\csc{{\left(\frac{{{3}\pi}}{{4}}\right)}}}+{\cot{{\left(\frac{{{3}\pi}}{{4}}\right)}}}\right|}-{\left(-{\ln}{\left|{\csc{{\left(\frac{\pi}{{4}}\right)}}}+{\cot{{\left(\frac{\pi}{{4}}\right)}}}\right|}\right)}$

$\displaystyle{L}=-{\ln}{\left|\sqrt{{{2}}}+{\left(-{1}\right)}\right|}-{\left(-{\ln{{\left(\sqrt{{{2}}}+{1}{\mid}\right)}}}=-{\ln}{\left|\sqrt{{{2}}}-{1}\right|}+{\ln}{\left|\sqrt{{{2}}}+{1}\right|}\right.}$

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

$\displaystyle{L}={\ln}{\left|\sqrt{{{2}}}+{1}\right|}-{\ln}{\left|\sqrt{{{2}}}-{1}\right|}={\ln}{\left|\frac{{\sqrt{{{2}}}+{1}}}{{\sqrt{{{2}}}-{1}}}\right|}$

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

$\displaystyle{L}={\ln}{\left|\frac{{{\left(\sqrt{{{2}}}+{1}\right)}}^{{2}}}{{1}}\right|}={\ln}{\left|{{\left(\sqrt{{{2}}}+{1}\right)}}^{{2}}\right|}={2}{\ln}{\left|\sqrt{{{2}}}+{1}\right|}$

So the exact value of the arc length of the graph of the function over the given interval is $\displaystyle{2}{\ln}{\left|\sqrt{{{2}}}+{1}\right|}$

which is approximately equal to 1.76.

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Friday, February 4, 2011

$\displaystyle{y}={\ln{{\left({\sin{{x}}}\right)}}},{\left[\frac{\pi}{{4}},\frac{{{3}\pi}}{{4}}\right]}$ Find the arc length of the graph of the function over the indicated interval.

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Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Friday, February 4, 2011

Find the arc length of the graph of the function over the indicated interval.

No comments:

Post a Comment

Thomas Jefferson&#39;s election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

$\displaystyle{y}={\ln{{\left({\sin{{x}}}\right)}}},{\left[\frac{\pi}{{4}},\frac{{{3}\pi}}{{4}}\right]}$ Find the arc length of the graph of the function over the indicated interval.

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?