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Numberical integration method

  1. Nov 28, 2014 #1
    1. The problem statement, all variables and given/known data

    Integreate:

    ##T = ∫ \frac{dy}{V_ab (y)} = \frac{2}{v}∫[1 + \frac{\alpha^2 y}{L} + 2\alpha \sqrt\frac{y}{L} cos(\phi(y))]^\frac{-1}{2} dy##

    where ## \phi (y) = \frac{\pi}{6} + sin^-1(\frac{\alpha\sqrt{y}}{2\sqrt{L}}) ##

    The limits are between 0 and L

    2. Relevant equations


    3. The attempt at a solution
    I have input this integral several times into matlab with no success, I was wondering if there was a way to do this on paper? My module on numerical integration isn't until next term
     
    Last edited: Nov 28, 2014
  2. jcsd
  3. Nov 28, 2014 #2

    Zondrina

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    Are you familiar with a function known as ##\text{quad()}##? It uses Simpson quadrature to numerically estimate the integral.

    There is also ##\text{quadl()}##, which uses Lobatto quadrature.

    The quad/quadl syntax you should be using is: ##\text{quad(function, a, b)}##.

    So define your integral as an anonymous function between the respective limits from ##a## to ##b## and quad should return the answer.
     
  4. Nov 28, 2014 #3

    Ray Vickson

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    You can use the fact that
    [tex] \cos(\phi) =\frac{\sqrt{3}}{4} \sqrt{4 - \frac{\alpha^2 y}{L}}- \frac{\alpha}{4} \sqrt{\frac{y}{L}}[/tex]
    then change variables to ##y/L = w^2## to get
    [tex]T = \frac{2}{v} 4L \int_0^1 \frac{w}{\sqrt{D(w)}} \, dw, \\
    D(w) = 4 + 2 \alpha^2 w^2 + 2 \sqrt{3} \alpha w \sqrt{4 - \alpha^2 w^2}
    [/tex]
    A further change of variables to ##w = (2/\alpha) \sin(\theta)##, followed by ##\theta = \lambda/2## produces
    [tex] T = \frac{2}{v} \frac{2L}{\alpha^2}
    \int_{\lambda =0}^{2\arcsin(\alpha/2)} \frac{\sin(\lambda)}{2 - \cos(\lambda) -\sqrt{3} \sin(\lambda)}\, d \lambda [/tex]
    This last integral still might not be doable explicitly, but it contains a single parameter ##r = 2 \arcsin(\alpha/2)##, so can be tabulated as a numerical function of ##r## (and perhaps even be "fitted" by a simple functional form in ##r## that has adequate accuracy over the ##r##-range of interest to you).

    Note added in edit: by some further manipulations, the integral can be done in terms of standard functions. I really cannot tell you more until you supply evidence of having struggled with the problem, by showing your work, etc.
     
    Last edited: Nov 28, 2014
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