The Proof is Trivial!

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)

Bumping with hints:

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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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)

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.

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.
$\displaystyle \frac{4a}{\pi} \int_0^{\pi} {(\pi - \theta) sin^2 \left( \frac{\theta}{2} \right) cos \left( \frac{\theta}{2} \right) } d\theta = \frac{32a}{9 \pi}$

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?

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?

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:

$E(X) = Var(X) = \lambda$

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?

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)