1. Integral Problem Help

The problem is y=x^(2/3) and x=1 and x=8. I need to find the exact arc length of the curve over the interval.

The formula used is the arc length formula: L=integral from a to b of (1+(dy/dx)^2) ^(1/2)

So far I got this:

Integral from 1 to 8 of (1+(4/9*x^(-2/3)))^(1/2)dx

I don't know how to integrate this. Is it a u substitution? I tried u= 1+ 4/9x^(-2/3) and I have an x^(-5/3) left over so that won't work.

2. Originally Posted by Simon777
The problem is y=x^(2/3) and x=1 and x=8. I need to find the exact arc length of the curve over the interval.

The formula used is the arc length formula: L=integral from a to b of (1+(dy/dx)^2) ^(1/2)

So far I got this:

Integral from 1 to 8 of (1+(4/9*x^(-2/3)))^(1/2)dx

I don't know how to integrate this. Is it a u substitution? I tried u= 1+ 4/9x^(-2/3) and I have an x^(-5/3) left over so that won't work.
For:
$\displaystyle \int_1^8 \sqrt{\frac{4}{9x^{\frac{2}{3}}}+1} dx$.

Make a common denominator to get:

$\displaystyle \int_1^8 \sqrt{ \frac{ 9x^{\frac{2}{3}}+4 }{ 9x^{\frac{2}{3} }} }dx$.

Now, Apply : $\displaystyle \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ for $\displaystyle b \neq 0$.

3. Originally Posted by Simon777
The problem is y=x^(2/3) and x=1 and x=8. I need to find the exact arc length of the curve over the interval.

The formula used is the arc length formula: L=integral from a to b of (1+(dy/dx)^2) ^(1/2)

So far I got this:

Integral from 1 to 8 of (1+(4/9*x^(-2/3)))^(1/2)dx

I don't know how to integrate this. Is it a u substitution? I tried u= 1+ 4/9x^(-2/3) and I have an x^(-5/3) left over so that won't work.
$\displaystyle f'(x) = \frac{2}{3}x^{-\frac{1}{3}}$

$\displaystyle f'(x) = \frac{2}{3x^{\frac{1}{3}}}$

$\displaystyle [f'(x)]^2 = \frac{4}{9x^{\frac{2}{3}}}$

$\displaystyle 1 + [f'(x)]^2 = 1 + \frac{4}{9x^{\frac{2}{3}}} = \frac{9x^{\frac{2}{3}}+4}{9x^{\frac{2}{3}}}$

$\displaystyle \sqrt{\frac{9x^{\frac{2}{3}}+4}{9x^{\frac{2}{3}}}} = \frac{1}{3x^{\frac{1}{3}}} \cdot \sqrt{9x^{\frac{2}{3}}+4}$

$\displaystyle \int_1^8 \frac{1}{3x^{\frac{1}{3}}} \cdot \sqrt{9x^{\frac{2}{3}}+4} \, dx$

let $\displaystyle u = 9x^{\frac{2}{3}}+4$

$\displaystyle du = \frac{6}{x^{\frac{1}{3}}} \, dx$

$\displaystyle \frac{1}{18} \int_{13}^{40} \sqrt{u} \, du$

can you finish ?

4. Originally Posted by Ted
For:
$\displaystyle \int_1^8 \sqrt{\frac{4}{9x^{\frac{2}{3}}}+1} dx$.

Make a common denominator to get:

$\displaystyle \int_1^8 \sqrt{ \frac{ 9x^{\frac{2}{3}}+4 }{ 9x^{\frac{2}{3} }} }dx$.

Now, Apply : $\displaystyle \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ for $\displaystyle b \neq 0$.

Wow, such a simple solution, I don't know why I didn't think to do that. Thank you so much for your help.

5. Originally Posted by skeeter
$\displaystyle f'(x) = \frac{2}{3}x^{-\frac{1}{3}}$

$\displaystyle f'(x) = \frac{2}{3x^{\frac{1}{3}}}$

$\displaystyle [f'(x)]^2 = \frac{4}{9x^{\frac{2}{3}}}$

$\displaystyle 1 + [f'(x)]^2 = 1 + \frac{4}{9x^{\frac{2}{3}}} = \frac{9x^{\frac{2}{3}}+4}{9x^{\frac{2}{3}}}$

$\displaystyle \sqrt{\frac{9x^{\frac{2}{3}}+4}{9x^{\frac{2}{3}}}} = \frac{1}{3x^{\frac{1}{3}}} \cdot \sqrt{9x^{\frac{2}{3}}+4}$

$\displaystyle \int_1^8 \frac{1}{3x^{\frac{1}{3}}} \cdot \sqrt{9x^{\frac{2}{3}}+4} \, dx$

let $\displaystyle u = 9x^{\frac{2}{3}}+4$

$\displaystyle du = \frac{6}{x^{\frac{1}{3}}} \, dx$

$\displaystyle \frac{1}{18} \int_{13}^{40} \sqrt{u} \, du$

can you finish ?
Thank you for working it out so I have something to check my steps with. Yes, I finished and checked my answer with the back of my book and it is correct.