Results 1 to 6 of 6

Math Help - Closed Curve Integral

  1. #1
    Senior Member bugatti79's Avatar
    Joined
    Jul 2010
    Posts
    461

    Closed Curve Integral

    Folks,

    Find a simple closed curve with counter clockwise rotation that maximizes the value of \int_{C} \frac{1}{3} y^3 dx + (x-\frac{1}{3} x^3) dx

    If I apply greens theoremI calculate \int \int_R (1-x^2-y^2) dA= \int_{0}^{2 \pi} \int_{0}^{1} (1-r^2)r dr d \theta = \frac{1}{2}...?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Dec 2011
    Posts
    83
    Thanks
    7

    Re: Closed Curve Integral

    Everything looks good except your answer. That integral does not evaluate to 1/2.

    You want to maximize 2\pi \int_a^b f'(r)dr=2\pi(f(b)-f(a)) where f'(r)=(1-r^2)r. Setting f'(r)=0 reveals where f has its min and max.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member bugatti79's Avatar
    Joined
    Jul 2010
    Posts
    461

    Re: Closed Curve Integral

    Quote Originally Posted by Ridley View Post
    Everything looks good except your answer. That integral does not evaluate to 1/2.

    You want to maximize 2\pi \int_a^b f'(r)dr=2\pi(f(b)-f(a)) where f'(r)=(1-r^2)r. Setting f'(r)=0 reveals where f has its min and max.
    Sorry, that answer is a typo. It should read pi / 2.

    How come we dont calculate a numerical answer even though the limits are specified..
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Closed Curve Integral

    Quote Originally Posted by bugatti79 View Post
    Folks,

    Find a simple closed curve with counter clockwise rotation that maximizes the value of \int_{C} \frac{1}{3} y^3 dx + (x-\frac{1}{3} x^3) dx

    If I apply greens theoremI calculate \int \int_R (1-x^2-y^2) dA= \int_{0}^{2 \pi} \int_{0}^{1} (1-r^2)r dr d \theta = \frac{1}{2}...?
    The request was to find the close curve C that maximizes the integral \int_{C} f(r)\ dr. In Your computation You suppose that C is the unity circle without supplying a prove of that...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Closed Curve Integral

    Because x^{2}+y^{2} is 'invariant to rotation' we can suppose that C is a circle centered in the origin and with radious R, so that we have to find the R that maximizes the contour integral. We can write...

    \int \int_{A} (1-x^{2}-y^{2})\ dx\ dy = 2\ \pi\ \int_{0}^{R} (1-r^{2})\ r\ dr (1)

    If we compute the derivative of the (1) respect to R we find that it vanishes for R\ (1-R^{2})=0 so that R=1 is the requested value and the curve is [effectively...] the unity circle...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member bugatti79's Avatar
    Joined
    Jul 2010
    Posts
    461

    Re: Closed Curve Integral

    Quote Originally Posted by chisigma View Post
    Because x^{2}+y^{2} is 'invariant to rotation' we can suppose that C is a circle centered in the origin and with radious R, so that we have to find the R that maximizes the contour integral. We can write...

    \int \int_{A} (1-x^{2}-y^{2})\ dx\ dy = 2\ \pi\ \int_{0}^{R} (1-r^{2})\ r\ dr (1)

    If we compute the derivative of the (1) respect to R we find that it vanishes for R\ (1-R^{2})=0 so that R=1 is the requested value and the curve is [effectively...] the unity circle...

    Kind regards

    \chi \sigma
    1) Looking at the double integral alone with the limits, without any concern for maximising...the answer is pi/2?

    2) Where did you get that term R(1-R^2)

    3) I find this conflicting with post #2..

    THanks
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: April 8th 2010, 04:58 PM
  2. integral, closed path
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: January 29th 2010, 01:58 PM
  3. closed curve, trace
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: January 27th 2010, 07:19 PM
  4. closed curve, trace
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: January 27th 2010, 06:59 PM
  5. Integral under a closed curve
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 24th 2009, 03:07 AM

Search Tags


/mathhelpforum @mathhelpforum