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    (Original post by atsruser)
    Problem 540 (**/***)

    a) Show that:

    \displaystyle \int_0^{2\pi} \frac{r^2\sin^2 2\theta}{1-2r^2\cos 2\theta+r^4} \ d\theta = \pi r^2

    stating any conditions that must hold for this to be valid.

    b) Find the value of \displaystyle \int_0^{2\pi} \frac{4\sin^2 2\theta}{17-8\cos 2\theta} \ d\theta

    (Hints available if necessary)
    (Original post by atsruser)
    Bumping with hints:

    Spoiler:
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    Hint 1)

    Note that since x^2+y^2=1, we have \int \frac{1}{\sqrt{1-x^2}} \ dx = \int \frac{dx}{y}.

    Which property of the curve does \frac{dx}{y} measure?

    Spoiler:
    Show

    dx is a fixed, small increment in x; how does \frac{dx}{y} vary as x varies from 0 to 1?; which property of the curve varies with dx in a similar way?

    Hint 2)

    To which points on the curve do the limits correspond? What is the significance of the values of the limits, and what is their relationship to the quantity measured by \frac{dx}{y}?

    Nice ones ... I spent 5 minutes on each and beat me.
    I will be back ...
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    (Original post by Zacken)
    Sorry to be a buzzkill, but it's specified in the OP to Latex all solutions/problems (I understand that you're on an iPad, but it's not hard to use Latex) - just to maintain austerity.

    Also - it's fine to post a solution to your own problem once somebody else has already answered it, but not before.
    I need to learn how to use it then. What if i just post a picture of the solution. I have some great problems with me that i would like to share. Also how do i spoiler things again?


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    (Original post by physicsmaths)
    I need to learn how to use it then. What if i just post a picture of the solution. I have some great problems with me that i would like to share. Also how do i spoiler things again?


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    Latex guide.

    If possible, problems in Latex and solutions in pictures works out fine!

    Eh, you don't really need to spoiler your answers, but you can spoiler things by using [spoiler]spoiler thing[ /spoiler] without a space in the last tag.
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    cheers



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    (Original post by Zacken)
    Eh, you don't really need to spoiler your answers, but you can spoiler things by using [spoiler]spoiler thing[ /spoiler] without a space in the last tag.
    And when you want to be able to demonstrate how to use a tag and don't want to have to put a space in to stop the tag working, you can use [noparse]. e.g.

    [noparse][spoiler]spoiler thing[/spoiler][/noparse]

    will look like:

    [spoiler]spoiler thing[/spoiler]
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    (Original post by DFranklin)
    x
    Ah, thanks very much for that! I was looking for the 'preformatted text' option in the toolbar but didn't see it.
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    (Original post by atsruser)
    Problem 540 (**/***)

    a) Show that:

    \displaystyle \int_0^{2\pi} \frac{r^2\sin^2 2\theta}{1-2r^2\cos 2\theta+r^4} \ d\theta = \pi r^2

    stating any conditions that must hold for this to be valid.

    b) Find the value of \displaystyle \int_0^{2\pi} \frac{4\sin^2 2\theta}{17-8\cos 2\theta} \ d\theta

    (Hints available if necessary)
    Hints:
    Spoiler:
    Show

    1. Is there a factorisation of 1-2r^2\cos 2\theta+r^4?
    Spoiler:
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    Maybe try 1-2r^2\cos 2\theta+r^4 = (1-r^2\alpha)(1-r^2\beta)
    Spoiler:
    Show
    Is there a nicer representation of that factorisation?
    Spoiler:
    Show
    Is this a complex problem?

    2. It may help to note that \sin^2 2\theta = \sin 2\theta \sin2\theta

    3. Would it help to consider another similar integral?
    Spoiler:
    Show
    with the same denominator, different numerator?
    Spoiler:
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    What about \int_0^{2\pi} \frac{1-r^2\cos 2\theta}{1-2r^2\cos 2\theta+r^4} \ d\theta?

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    Problem 542 (*/**)

    1. A line is of length L. Point P lies l distant from one end of the line. Find the average distance from P to all other points on the line.

    2. A disc is of radius a. Find the average distance from the centre of the disc to all other points in the disc.

    3. For the same disc, find the average distance from a fixed point P on the circumference to all other points in the disc.
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    (Original post by atsruser)
    Problem 542 (*/**)
    2. A disc is of radius a. Find the average distance from the centre of the disc to all other points in the disc.
    Roughly speaking, is this equivalent to finding the distance of one independently chose point on the unit disk. I've heard something similar about the mean distance between points on a disk, does it apply to the distance between points on a disk and it's centre?
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    (Original post by atsruser)
    Problem 542 (*/**)

    1. A line is of length L. Point P lies l distant from one end of the line. Find the average distance from P to all other points on the line.

    2. A disc is of radius a. Find the average distance from the centre of the disc to all other points in the disc.

    3. For the same disc, find the average distance from a fixed point P on the circumference to all other points in the disc.
    For part 3.
    \displaystyle \frac{4a}{\pi} \int_0^{\pi} {(\pi - \theta) sin^2 \left( \frac{\theta}{2} \right) cos \left( \frac{\theta}{2} \right) } d\theta = \frac{32a}{9 \pi}
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    (Original post by EricPiphany)
    For part 3.
    \displaystyle \frac{4a}{\pi} \int_0^{\pi} {(\pi - \theta) sin^2 \left( \frac{\theta}{2} \right) cos \left( \frac{\theta}{2} \right) } d\theta = \frac{32a}{9 \pi}
    That looks to be correct, but bear in mind that the value of these questions isn't in knowing the answer, it's in seeing the argument, and you haven't given one. For example, I approached this by doing a double integral, which is easy to justify; how did you write down that single integral?
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    (Original post by Zacken)
    Roughly speaking, is this equivalent to finding the distance of one independently chose point on the unit disk.
    I'm not sure what you're asking here: the distance of what from what? It looks like you are asking about some kind of probabilistic question, but this isn't a problem in probability (which is good, for probability is the work of Beelzebub)

    I've heard something similar about the mean distance between points on a disk, does it apply to the distance between points on a disk and it's centre?
    Again you have me baffled here. You haven't told me what the "something similar" is, so I can't tell if it applies to anything, I'm afraid.
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    (Original post by atsruser)
    That looks to be correct, but bear in mind that the value of these questions isn't in knowing the answer, it's in seeing the argument, and you haven't given one. For example, I approached this by doing a double integral, which is easy to justify; how did you write down that single integral?
    Let P be a point on the circumference of the disk, and O be the center of the disk.
    Draw a circle around P intersecting the edge of the disk at S and  T, let \theta = \angle POS, r = |PS| and \phi = \angle SPT = \pi - \theta.
    Now let another slightly larger concentric circle be drawn around P intersecting the edge of the disk at U such that \angle POU = \theta + \delta \theta, the thickness of the resulting shell being \delta t.
    Now we have r = 2a sin \left( \frac{\theta}{2} \right) and \delta t \approx a cos \left( \frac{\theta}{2} \right) \delta \theta and arc length  \arc ST = r \phi .

    Someone finish this proof for me, I can't type any more Latex on my phone.
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    Problem 543 *

    Since this thread has been a bit quiet I'll post a relatively easy question which would have made OCR MEI'S S2 a lot more interesting!

    Let the DRV X follow a Poisson distribution with parameter \lambda Prove that:

     E(X) = Var(X) = \lambda
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    (Original post by 16Characters....)
    Problem 543 *

    Since this thread has been a bit quiet I'll post a relatively easy question which would have made OCR MEI'S S2 a lot more interesting!

    Let the DRV X follow a Poisson distribution with parameter \lambda Prove that:

     E(X) = Var(X) = \lambda
    Solution 543:
    Since generating functions are a bit of a faff I'll do it directly.

    E(X):=\displaystyle\sum_{x \in Im(X)} xP(X=x)
    For a Poisson variable, this means that:
    E(X)= \displaystyle\sum_{k=0}^{\infty} k\dfrac{e^{-\lambda} \lambda ^k}{k!}=\lambda e^{-\lambda}\displaystyle\sum_{k=1}^  {\infty} \dfrac{\lambda ^{k-1}}{(k-1)!}= \lambda e^{-\lambda} \displaystyle\sum_{k=0}^{\infty}  \dfrac{\lambda ^k}{k!}

\Rightarrow E(X)=\lambda e^{-\lambda} \cdot e^{\lambda}=\lambda

    Var(X):=E(X^2)-(E(X))^2
    Observe that E(X^2-X)=E(X^2)-E(X) as expectation is a linear operator.
    Thus: \Var(X)=E(X^2-X)+E(X)-(E(X))^2.
    But:
    E(X^2-X)=\displaystyle\sum_{k=0}^{ \infty} \dfrac{(k^2-k)}{k!}e^{-\lambda} \lambda^k=\lambda ^2 e^{-\lambda} \displaystyle\sum_{k=2}^{\infty} \dfrac{\lambda ^{k-2}}{(k-2)!}=\lambda^2 e^{-\lambda} \displaystyle\sum_{k=0}^{\infty} \dfrac{\lambda^{k}}{k!}.

    \Rightarrow E(X^2-X) = \lambda^2

 e^{-\lambda} \cdot e^{\lambda}=\lambda ^2
    Thus: Var(X)=(\lambda^2+\lambda)-\lambda^2=\lambda.

    Hence E(X) = Var(X)= \lambda.  \ \ \ \ \Box
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    Problem 544***

    Show that

    \displaystyle \int_{0}^{2\pi} \dfrac{\cos 3x}{5 - 4\cos x} \ dx = \dfrac{\pi}{12}
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    (Original post by Indeterminate)
    Problem 544***

    Show that

    \displaystyle \int_{0}^{2\pi} \dfrac{\cos 3x}{5 - 4\cos x} \ dx = \dfrac{\pi}{12}
    I've done this, via a contour integral. However, calculating one of the residues was messy (at least the way I did it) so I'm wondering if there's some slicker approach that you had in mind. Any hints?
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    (Original post by atsruser)
    I've done this, via a contour integral. However, calculating one of the residues was messy (at least the way I did it) so I'm wondering if there's some slicker approach that you had in mind. Any hints?
    I haven't tried it myself, but perhaps working with \dfrac{1}{5 - 4\cos x} and \cos x in its exponential form and then partial fractions and geometric series may prove useful.
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    (Original post by Zacken)
    I haven't tried it myself, but perhaps working with \dfrac{1}{5 - 4\cos x} and \cos x in its exponential form and then partial fractions and geometric series may prove useful.
    Would you like to expand on that a bit? Not sure I see where the partial fractions and geometric series will arise. (Though if you can end up with a series in \cos nx that will be very useful)
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    (Original post by atsruser)
    Would you like to expand on that a bit? Not sure I see where the partial fractions and geometric series will arise. (Though if you can end up with a series in \cos nx that will be very useful)
    Maybe he is talking about doing a GP on the reals of tbe imaginary numbers. I have had a try and came up as problematic as i found cos3x(pretty easy) 4cos^3(x)-3cosx now divide this to get a quadratic in cosx and the fraction A/(5-4cosx) problem is i get the integral of that through u=tan(x/2) sub but the limits are problematic as they are both zero it seems.


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