Results 1 to 7 of 7
Like Tree2Thanks
  • 1 Post By HallsofIvy
  • 1 Post By Prove It

Math Help - Exact length of a curve?

  1. #1
    Member
    Joined
    Dec 2012
    From
    NY
    Posts
    105
    Thanks
    1

    Exact length of a curve?

    Can anyone explain a method on how to go about this problem? I don't care for the answer; I'm just perplexed on how to begin the problem in a successful manner. Thanks!



    edit here's syntax:
    Last edited by Steelers72; April 24th 2013 at 11:08 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,579
    Thanks
    1418

    Re: Exact length of a curve?

    Where did you get this problem? It is pretty standard Calculus problem- are you not taking Calculus and have not learned this?

    If y= f(x), the arclength, from x= a to x= b, is given by \int_a^b \sqrt{1+ f'^2(x)} dx.
    Thanks from Steelers72
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Dec 2012
    From
    NY
    Posts
    105
    Thanks
    1

    Re: Exact length of a curve?

    Thanks! Oh, so you have to find the derivative of y and plug into the length formula you've provided right?

    My instructor hasn't covered this (yet?) and gave this as one of a bunch of questions to do for next week. Had no clue what formula to use but thanks for clearing that up!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Dec 2012
    From
    NY
    Posts
    105
    Thanks
    1

    Re: Exact length of a curve?

    By the way, how would you go about finding the intervals? I'm guessing set y=0? Not sure.



    I've gotten this far:

    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Dec 2012
    From
    NY
    Posts
    105
    Thanks
    1

    Re: Exact length of a curve?

    Original problem:y=sqrt(x-x^2)+arcsin(sqrt(x))

    Sorry, this is the derivative of y I got to plug into the Length formula:

    sqrt(-(-1+x) x)/x


    Does anyone know what the bounds are and how to find them? Do we set the original y equation = to 0?

    I think it may be 0 to 1 but have no clue as to why.
    Last edited by Steelers72; April 25th 2013 at 02:37 PM.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,513
    Thanks
    1404

    Re: Exact length of a curve?

    Quote Originally Posted by Steelers72 View Post
    Original problem:y=sqrt(x-x^2)+arcsin(sqrt(x))

    Sorry, this is the derivative of y I got to plug into the Length formula:

    sqrt(-(-1+x) x)/x


    Does anyone know what the bounds are and how to find them? Do we set the original y equation = to 0?

    I think it may be 0 to 1 but have no clue as to why.
    \displaystyle \begin{align*} f(x) &= \sqrt{ x - x^2 } + \arcsin{ \left( \sqrt{x} \right) } \\ \\ f'(x) &= \frac{1 - 2x}{2\,\sqrt{ x - x^2 }} + \frac{1}{2\,\sqrt{x} \, \sqrt{  1 - x } } \\ &= \frac{ 1 - 2x }{ 2 \, \sqrt{ x \left( 1 - x \right) } } + \frac{1}{2 \, \sqrt{ x \left( 1 - x \right) } } \\ &= \frac{2 - 2x}{2\,\sqrt{x \left( 1 - x \right) }} \\ &= \frac{1 -x}{\sqrt{x \left( 1 - x \right) }} \\ &= \frac{ \sqrt{x \left( 1 - x \right) } }{x} \end{align*}

    So now plugging into the formula for the arclength (I can't see your images so I'm not sure what your bounds should be)...

    \displaystyle \begin{align*} l &= \int_a^b{ \sqrt{ 1 + \left[ f'(x) \right] ^2 } \, dx } \\ &= \int_a^b{ \sqrt{1 + \left[ \frac{\sqrt{x\left( 1 - x \right) }}{x} \right] ^2 } \, dx } \\ &= \int_a^b{ \sqrt{ 1 + \frac{x \left( 1 - x \right) }{x^2} } \,dx } \\ &= \int_a^b{ \sqrt{ 1 + \frac{1 - x}{x} }\,dx } \\ &= \int_a^b{ \sqrt{ \frac{1}{x} } \,dx  } \\ &= \int_a^b{ x^{-\frac{1}{2} } \,dx} \\ &= \left[ 2x^{\frac{1}{2}} \right] _a^b \\ &= 2\,\sqrt{b} - 2\,\sqrt{a} \end{align*}

    So now plug in your endpoints...
    Thanks from Steelers72
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Dec 2012
    From
    NY
    Posts
    105
    Thanks
    1

    Re: Exact length of a curve?

    Thanks for the work through! I was confused as to how to get those endpoints but your work through helps a lot too.

    Edit: never mind. They were 0 to 1
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Determining an exact expression for a length.
    Posted in the Trigonometry Forum
    Replies: 5
    Last Post: December 6th 2009, 02:14 PM
  2. exact length of curve
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 8th 2009, 10:38 PM
  3. Help me ! Find the exact length of the curve
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 25th 2009, 11:32 AM
  4. arc length and parameterized curve length
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 5th 2008, 02:33 AM
  5. help finding exact arc length of a curve
    Posted in the Calculus Forum
    Replies: 5
    Last Post: December 2nd 2008, 04:10 PM

Search Tags


/mathhelpforum @mathhelpforum