Math Help - Find the length of the curve

1. [SOLVED] Find the length of the curve

Find the exact length of the curve defined by 0= x^3 - 8xy + 3 from (1,(2/6)) to (3, (10/6)) .

How is this problem to be approached. Do I turn the given into a linear equation and use the given x values in a definate integral?
Not sure what to do.

Your guidance is appreciated.

2. Re: Find the length of the curve

The length of the curve is found as $\displaystyle L = \int_a^b\sqrt{1+\left(\frac{dy}{dx}\right)^2}$

Notice your equation becomes $\displaystyle y = \frac{1}{8}\left(x^2+\frac{3}{x}\right)$

3. Re: Find the length of the curve

Ah. So in order to find this length, you would take the derivative of the rearranged equation and plug it into the formula for "L" yes? That much I understand. My concern lies determining a and b using the givens.

4. Re: Find the length of the curve

Originally Posted by andvaka
Ah. So in order to find this length, you would take the derivative of the rearranged equation and plug it into the formula for "L" yes? That much I understand. My concern lies determining a and b using the givens.
$L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$

$a$ and $b$ are x-values of the curve's end points

$L = \int_c^d \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy$

$c$ and $d$ are y-values of the curve's end points

5. Re: Find the length of the curve

Originally Posted by andvaka
Find the exact length of the curve defined by 0= x^3 - 8xy + 3 from (1,(2/3)) to (3, (14/3)).
Those points do not lie on that curve, so I really don't understand the question.

6. Re: Find the length of the curve

Originally Posted by andvaka
Find the exact length of the curve defined by 0= x^3 - 8xy + 3 from (1,(2/3)) to (3, (14/3)) .

How is this problem to be approached. Do I turn the given into a linear equation and use the given x values in a definate integral?
Not sure what to do.

Your guidance is appreciated.
I recommend you go back and read the problem again. As reckoner said, the points you give, (1,(2/3)) and (3, (14/3)) are NOT on the given curve, $x^3 - 8xy + 3= 0$. If x= 1 and y= 2/3, then $x^3- 8xy+ 3= 1- 16/3+ 3= -4/3$, not 0. And if x= 3, y= 14/3, then $x^3- xy+ 3= 27- 112+ 3= -82$, not 0.

7. Re: Find the length of the curve

I can read the problem again, but that wouldn't bring me any close to a solution would it? The problem is exactly as I stated it. I'm just trying the most constructive way to approach this. I suppose I will have to ask my calculus professor to elaborate so that I know what it is she wants me to do here.

8. Re: Find the length of the curve

Have you found $\frac{dy}{dx}$ ?

I think you should ask your professor if these points are supposed to be on the curve? You can illustrate they are not.