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Area bounded by a curve's loop

  1. Apr 27, 2015 #1
    1. The problem statement, all variables and given/known data
    The area bounded by the loop of the curve ## 4y^2 = x^2(4-x^2) ## is in sq. units
    7/3
    8/3
    11/3
    16/3

    2. Relevant equations
    NA

    3. The attempt at a solution
    By putting x = 0 and x = 2 I am getting y = 0.
    Getting complex y values after x exceeds 2.
    I am not getting where the loop would form
    I am not allowed to use graphing calculator to see shape.
    Is there any better way than plotting for concavity of function, to determine loop, or something else?
     
  2. jcsd
  3. Apr 27, 2015 #2

    HallsofIvy

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  4. Apr 27, 2015 #3

    Zondrina

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    Start by finding a polar representation of the given curve. You wind up with something like this:

    $$r = f(\theta) = \sqrt{4\text{sec}^2(\theta) (1 - \text{tan}^2(\theta))}$$

    Now you have a theorem which tells you the area of a polar region ##R## is given by:

    $$\frac{1}{2} \int_a^b f^2(\theta) \space d \theta$$
     
  5. Apr 27, 2015 #4
    But you may have used graphing calculator to see the shape as a lemniscate.
    How to know shape if you have not used the calculator?
     
  6. Apr 27, 2015 #5
    I think for finding area once we know graph can easily do,
    $$ \int_0^2 \sqrt{x^2(4-x^2)}dx $$
     
  7. Apr 27, 2015 #6

    Zondrina

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    There are several ways to graph ##r = f(\theta)##. The one I usually use is to plot ##f(\theta)## in the Cartesian plane for several radial arguments. This will allow you to trace the graph in the ##r - \theta## plane and obtain the limits of integration. These are the limits for the theorem in post #3.

    The theorem actually makes it quite easy to integrate in polar co-ordinates as well (which is what the exercise intended I believe). The substitution ##u = \text{tan}^2(\theta)## makes the rest easy.
     
    Last edited: Apr 27, 2015
  8. Apr 27, 2015 #7

    Ray Vickson

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    Yes, but the OP's alternative ##\int \sqrt{x^2(4-x^2)} \, dx = \int x \sqrt{4 - x^2} \, dx## is even easier.
     
  9. Apr 27, 2015 #8
    I have not read that polar coordinate things till now.
    Can you tell me the answer for below quotation?
     
  10. Apr 27, 2015 #9

    Ray Vickson

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    See, eg., http://en.wikipedia.org/wiki/Lemniscate . Click on the link to "Lemniscate of Gerono".
     
  11. Apr 27, 2015 #10
    But suppose we do not know these things.
    Doing some basics. We have to find only area of any curve . For example here I found y value 0 when x is 0 and y value 0 when x is 2. I am getting complex number value when taking x value more than 2.
    From this how I can arrive at shape?
     
  12. Apr 27, 2015 #11

    Ray Vickson

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    Plot the curve.

    Do not tell me that you are not allowed to plot things; of course you are---you just cannot turn in the plot as a final solution for mark! Nobody can stop you from making plots in the privacy of your own room. The plot does not have to be very accurate; it is enough that it supplies a rough idea of the shape.

    Alternatively: to see what ##y = c x \sqrt{4 - x^2}## looks like look at ##y^2 = c^2 x^2(4-x^2)##. As a function of ##t = x^2##, the rhs is ##c^2 t(4-t)##, which rises from 0 at ##t = 0## to a maximum at ##t = 2## and then falls to 0 again at ##t = 4##. So ##y^2## rises fro 0 to a maximum at ##x = \sqrt{2}##, then falls to 0 again at ##x = 2##. Wherever ##y^2## is rising, so is ##y##, and wherever ##y^2## is falling, so is ##y##. Therefore, ##y## rises from 0 to a maximum at ##x = \sqrt{2}## then falls to zero again.

    The lower branch ##y = - c x \sqrt{4 - x^2}## falls from 0 to a minimum then rises again to 0.
     
  13. Apr 27, 2015 #12

    Zondrina

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    Really though, polar co-ordinates might help you plot this easier. You shouldn't be lazy about doing plots, they are tedious, but a necessity.

    A picture is worth a million words.

    Speaking of pictures, maybe this one will help you understand what I meant in post #6:

    Screen Shot 2015-04-27 at 4.49.32 PM.png

    The function ##r = f(\theta)## might look simpler, but the procedure is the same.
     
  14. Apr 28, 2015 #13
    Thanks Zondrina, Ray, Halls although Zondrina I don't know anything about polar coordinate system.
    In school level I suppose it is not taught.
     
  15. Apr 28, 2015 #14
    It is usually taught in high school. Maybe your teacher thought it was too obvious to mention. See here.
    [EDIT:- you just need to know what ##r## and ##\theta## (and ##\phi## if your working in 3 dimensions) are, that's all there is to it really.]
     
  16. Apr 28, 2015 #15
    What is the use of polar coordinates when we can do the thing of post 5 easily?
     
  17. Apr 28, 2015 #16
    Nothing. But polar coordinates become extremely useful when you try to describe curves like cycloids and trochoids. A lot of their properties become a lot easier to prove, describe and visualize in polar coordinates. I'm not sure what level you're at, but the 4th chapter of Courant's analysis book has an excellent section on the theory of plane curves. If you read it, the use of polar coordinates will become immediately clear.
     
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