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#1 Help Me ! » Triple integral from hell... » 2007-02-03 23:35:00

asv
Replies: 1

Hi all,

I have to evaluate the volume integral ∫∫∫ ( 2 - y²/x² - x²/y² ) dV over the circular cylinder bounded by the planes z=0 and z=1 and the curved surface x²+y²=a² for 0≤z≤1.

The approach I've tried was to integrate w.r.t z first, leaving ∫∫ ( 2 - y²/x² - x²/y² ) dxdy, but that didn't end up getting anywhere because the limits for x and y left things that seemed fairly unintegratable.

Then I tried going polar, giving me ∫∫∫ ( 2 - tan²θ - cot²θ ) dθdrdz = ∫∫ ( 2 - tan²θ - cot²θ ) drdθ. This seems to come out as [4θ-tanθ+cotθ] evaluated between 2pi and 0 (?), but cotθ isn't defined at these values :S

So yeah, I'm kinda stuck dizzy - any help would be greatly appreciated smile

asv smile

#2 Re: Help Me ! » Simple Fourier Series Question » 2006-10-11 21:12:44

asv

Thanks for your reply.

I can't get 1/2 for the value of bn. Here's my approach:

pi*bn=∫f(x)sin(nx)dx= ∫sin(x)sin(nx) dx [evaluated between 0 and pi]


=1/2 ∫ (-cos((1+n)x) + cos((1-n)x)) dx  {using that 2sinAsinB=-cos(A+B)+cos(A-B)}

=1/2 [  (-sin((n+1)x))/(n+1) + (sin((n-1)x))/(n-1) ] evaluated between 0 and pi

and upon substituting the limits (with n a natural number), the sine of 0 and the sine of an integer multiple of pi are both zero, so the whole thing is 0.



Plus, if bn was 1/2, wouldn't the question say:


"Show that the Fourier series of f is:

1/pi + ∑(sin(nx))/2 - (2/pi)∑((cos(2nx)/(4n²-1))."  ?

(with the sums running from n=1 to infinity)

asv

EDIT: Please ignore me. Problem solved. Thanks for your help!

#3 Help Me ! » Simple Fourier Series Question » 2006-10-11 09:49:53

asv
Replies: 3

Hi all,

Here's the question:

---

f(x)= 0 if -pi≤x≤0
     = sin x if 0<x<pi

Show that the Fourier series of f is:

1/pi + (sinx)/2 - (2/pi)∑((cos(2nx)/(4n²-1)).

---

Now, here's my problem...
I can get the 1/pi, i can get where the -(2/pi)∑((cos(2nx)/(4n²-1)) comes from. But where on earth is the (sinx)/2 from?

Please help!

asv

#4 Help Me ! » Statistics - Unbiased Estimators and Standard Error » 2006-02-15 12:24:26

asv
Replies: 0

Hi all,

Just found this question and I'm not sure how to approach it - any help is greatly appreciated smile

---
Suggest an unbiased estimator of the unknown θ and evaluate its standard error:
Z has expected value 0, variance θ and E[Z^4]=2θ²
---

Thanks

#5 Help Me ! » A tricky problem to get you all thinking... » 2006-01-18 14:07:20

asv
Replies: 2

OK, this might seem quite a random problem but please have a think about this...

I was thinking, pi has been calculated to 1,241,100,000,000 decimal places. There are 6,670,903,752,021,072,936,960 possible valid sudoku grids. What is the probability that there are 81 consecutive digits that are a valid sudoku within the digits of pi we already know?

Hope that makes sense.

asv
(PS does anyone know a busier maths forum that might be better suited to problems like this?)

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