# Arc Length

• Jul 31st 2006, 01:33 PM
c_323_h
Arc Length
Given $y=f(x)$ the formula for arc length $L$ over the interval $[a,b]$:

$L = \int_a^b \sqrt{1+[f'(x)]^2} dx$
or $\int_a^b \sqrt{1+(\frac{dy}{dx})^2}dx$

With the formula above find the arc length of the following functions over the given interval.

1. $y=1+6x^{3/2}, [0,1]$
Answer: $\frac{2}{243}(82\sqrt{82}-1)$

2. $y=\frac{x^5}{6}+\frac{1}{10x^3}, [1,2]$
Answer: $\frac{1261}{240}$

I found the derivatives of the functions, $9\sqrt{x}$ and $\frac{5x^4}{6}-\frac{3}{10x^4}$, respectively, and plugged them in, simplified, substituted and converted limits if I had to, but still don't get the correct answer. Could someone tell me what I'm doing wrong?
• Jul 31st 2006, 01:41 PM
ThePerfectHacker
Quote:

Originally Posted by c_323_h

1. $y=1+6x^{3/2}, [1,2]$
Answer: $\frac{43}{3}$

You have,
$y=1+6x^{3/2}$
Thus,
$y'=6(3/2)x^{1/2}$
Simplfy,
$y'=9x^{1/2}$
Thus,
$\sqrt{1+[f'(x)]^2}=\sqrt{1+[9x^{1/2}]^2}=\sqrt{1+81x}$
Thus,
$\int_1^2 \sqrt{1+81x}dx$
Can you integrate that? Or do you need the quickest integrator in the west to help you?
• Jul 31st 2006, 01:50 PM
c_323_h
Quote:

Originally Posted by ThePerfectHacker
You have,
$y=1+6x^{3/2}$
Thus,
$y'=6(3/2)x^{1/2}$
Simplfy,
$y'=9x^{1/2}$
Thus,
$\sqrt{1+[f'(x)]^2}=\sqrt{1+[9x^{1/2}]^2}=\sqrt{1+81x}$
Thus,
$\int_1^2 \sqrt{1+81x}dx$
Can you integrate that? Or do you need the quickest integrator in the west to help you?

What I tried:

let $u=1+81x$, then $du=81dx$.
Convert limits: when $x=0, u=1$ and when $x=1, u=82$

so, $\frac{1}{81}\int_1^{82}\sqrt{u}du$
$\frac{1}{81}(\frac{2u^{3/2}}{3}) |_1^{82}$

substitute $u$ back in. Evaluate at limits using the Fundamental Theorem of Calculus. Is this correct?
• Jul 31st 2006, 02:38 PM
Soroban
Hello, c_323_h!

What you've done is correct . . .

Quote:

Let $u=1+81x$, then $du=81\,dx$
Convert limits: when $x=0, u=1$ and when $x=1, u=82$

So: $\frac{1}{81}\int_1^{82}\!\!\sqrt{u}\:du \;= \;\frac{1}{81}\left(\frac{2u^{3/2}}{3}\right) \bigg|_1^{82}\quad\Rightarrow\quad\frac{2}{243}u^{ \frac{3}{2}}\bigg|^{82}_1$

Evaulate: . $\frac{2}{243}\left(82^{\frac{3}{2}} - 1^{\frac{3}{2}}\right) \;= \;\frac{2}{243}\left(82\sqrt{82} - 1\right)$

And is there a typo in #2?
Is there a plus between the two fractions?

• Jul 31st 2006, 02:48 PM
c_323_h
Quote:

Originally Posted by Soroban

yup it's supposed to be a plus and the answers are indeed switched. i'll edit my post..also the answer to the second question is $\frac{1261}{240}$ and not $\frac{46}{3}$