Question

arclength of ln(sinx) from pi/4 to 3pi/4:

Answer 1

So I use the arc-length formula:
\[\int\limits_{\pi/4}^{3\pi/4}\sqrt{1+(\frac{d}{dx}\ln(\sin(x)))^2}dx\]

which gives me
\[\int\limits\limits_{\pi/4}^{3\pi/4}\sqrt{1+\cot^2(x)}dx\]
and after using the Pythagorean identity:
\[\int\limits\limits\limits_{\pi/4}^{3\pi/4}\csc(x)dx\]

NOW THEN integrating it:
\[-\ln|cscx+cotx| |_{\pi/4}^{3\pi/4}\]

And this turns out to be 0.

An arc-length of 0 is what throws me off, this can't be right...right?

Answer 2

\(L=-\ln|cscx+cotx| |_{\pi/4}^{3\pi/4}\)
\(L=-\ln|csc(\frac{3}{4}\pi)+cot(\frac{1}{4}\pi)|-(-\ln|csc(\frac{3}{4}\pi)+cot(\frac{1}{4}\pi)|)\)
\(L=-\ln|\sqrt{2}+(-1)|-(-\ln|\sqrt{2}+(+1)|)=-In|\sqrt{2}-1|+In|\sqrt{2}+1|\)
From here,switch the two natural log and use the quotient property to combine them into a single log:
\(L=In|\sqrt{2}+1|-In|\sqrt{2}-1|=In|\frac{(\sqrt{2}+1)}{(\sqrt{2}-1)}|\)
Then rationalize the denominator by multiplying the conjugate and simplifying and use the power property of logs,and you will get:
\(L=In|\frac{(\sqrt{2}+1)^2}{1}|=In|(\sqrt{2}+1)^2)|=2In|\sqrt{2}+1|\)

Answer 3

\(\approx1.76\)

Answer 4

Hmmm well..I'm confused about the transition from line 2 to line 3, where cot(pi/4) becomes +1 or -1.

Answer 5

Ops sorry 4 the confusion >.< , there is some typo error in line 2...

Answer 6

\(\color{#0cbb34}{\text{Originally Posted by}}\) @Logic007
\(L=-\ln|cscx+cotx| |_{\pi/4}^{3\pi/4}\)
\(L=-\ln|csc(\frac{3}{4}\pi)+\color{blue}{cot(\frac{1}{4}\pi)}|-(-\ln|csc(\frac{3}{4}\pi)+\color{blue}{cot(\frac{1}{4}\pi)}|)\)
\(L=-\ln|\sqrt{2}+(-1)|-(-\ln|\sqrt{2}+(+1)|)=-In|\sqrt{2}-1|+In|\sqrt{2}+1|\)
From here,switch the two natural log and use the quotient property to combine them into a single log:
\(L=In|\sqrt{2}+1|-In|\sqrt{2}-1|=In|\frac{(\sqrt{2}+1)}{(\sqrt{2}-1)}|\)
Then rationalize the denominator by multiplying the conjugate and simplifying and use the power property of logs,and you will get:
\(L=In|\frac{(\sqrt{2}+1)^2}{1}|=In|(\sqrt{2}+1)^2)|=2In|\sqrt{2}+1|\)
\(\color{#0cbb34}{\text{End of Quote}}\)
\(L=-\ln|csc(\frac{3}{4}\pi)+\color{red}{cot(\frac{3}{4}\pi)}|-(-\ln|csc(\frac{1}{4}\pi)+\color{red}{cot(\frac{1}{4}\pi)}|)\)

Answer 7

@mhchen

Answer 8

\(cot(\frac{\pi}{4})=+1\)

Answer 9

\(cot(\frac{3\pi}{4})=-1\)

Answer 10

omg you're right

dam so that's what I missed on my midterm xD Thanks for clarifying cause half of my class told me they got 0 on that question and we all missed it lol

Answer 11

Haha,no prob.
Thanks for asking. ^