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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 - any help would be greatly appreciated
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!
Hi all,
Here's the question:
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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
Hi all,
Just found this question and I'm not sure how to approach it - any help is greatly appreciated
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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
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Thanks
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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