Length between curve

• Nov 24th 2008, 05:56 PM
elitespart
Length between curve
I need to find the length of the curve y=(1/2)x^2 - (1/4)lnx between x=1 and x=8.

I derived it and squared it and added one to it so now I'm left with the integral from 1 to 8 of sqrt (x^2 + (1/2) + 1/16x^2) dx. How do I integrate this?
• Nov 24th 2008, 06:23 PM
o_O
$\displaystyle L = \int_1^8 \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \ dx$

Now, as you found: $\displaystyle \frac{dy}{dx} = x - \frac{1}{4x}$

So:
\displaystyle \begin{aligned} L & = \int_1^8 \sqrt{1 + \left(x - \frac{1}{4x}\right)^2} \ dx \\ & = \int_1^8 \sqrt{{\color{red}\left(\frac{x^2}{x^2}\right)}\le ft(x^2 + \frac{1}{16x^2} + \frac{1}{2}\right)} \ dx \\ & = \int_1^8 \sqrt{\frac{x^4 +\frac{1}{2}x^2 + \frac{1}{16}}{x^2}} \ dx \\ & = \int_1^8 \sqrt{\frac{\left(x + \frac{1}{4}\right)^2}{x^2}} \end{aligned}

So, taking the square root and our final integral becomes: $\displaystyle L = \int_1^8 \frac{x + \frac{1}{4}}{x} \ dx \ = \ \int_1^8 \left(1 + \frac{1}{4x}\right) dx$
• Nov 24th 2008, 06:23 PM
Soroban
Hello, elitespart!

Quote:

Find the length of the curve: .$\displaystyle y\:=\:\tfrac{1}{2}x^2 - \tfrac{1}{4}\ln x$ between $\displaystyle x=1$ and $\displaystyle x=8.$

I derived it and squared it and added one to it.

So now I'm left with: .$\displaystyle L \;=\;\int^8_1\sqrt{x^2 + \tfrac{1}{2} + \tfrac{1}{16x^2}}\,dx$
How do I integrate this?

Believe it or not, there is a perfect square under the radical . . .

. . $\displaystyle \sqrt{x^2 + \frac{1}{2} + \frac{1}{16x^2} }\;=\;\sqrt{\left(x + \frac{1}{4x}\right)^2} \;=\;x + \frac{1}{4x} \quad\hdots$ see?

• Nov 24th 2008, 06:26 PM
elitespart
Awesome! Thanks a lot guys.
• Nov 24th 2008, 06:38 PM
Krizalid
It's actually $\displaystyle \sqrt{\left( x+\frac{1}{4} \right)^{2}\cdot \frac{1}{x^{2}}}=\left| \left( x+\frac{1}{4} \right)\cdot \frac{1}{x} \right|,$ but note that $\displaystyle \left( x+\frac{1}{4} \right)\cdot \frac{1}{x}>0$ for $\displaystyle 1\le x\le8,$ hence $\displaystyle \left| \left( x+\frac{1}{4} \right)\cdot \frac{1}{x} \right|=\left( x+\frac{1}{4} \right)\cdot \frac{1}{x}.$