Any help with the following problem is appreciated (I have no idea how to approach it):

Consider the integral . Express as a double integral involving and .

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- May 1st 2010, 12:55 AMdemodeDouble Integration
Any help with the following problem is appreciated (I have no idea how to approach it):

Consider the integral . Express as a double integral involving and . - May 1st 2010, 12:58 AMProve It
- May 1st 2010, 01:06 AMdemode
- May 1st 2010, 01:13 AMProve It
- May 1st 2010, 03:33 AMHallsofIvy

since that is just the same integral with "dummy variable" y rather than x.

Then

by Fubini's theorem, that is the same as

- May 1st 2010, 04:27 AMProve It
- May 1st 2010, 07:13 PMdemode
- May 2nd 2010, 12:36 AMProve It
You know that

http://upload.wikimedia.org/math/d/6...1e85083865.png

.

You can also say

http://upload.wikimedia.org/math/f/1...7e77514b2f.png

Therefore

which means .

Another handy application is using the Gaussian Integral to compute .

Since is an even function, that means

.

Using the substitution yields

and

So that the integral becomes

.

So that means . - May 2nd 2010, 03:47 AMHallsofIvy
- May 2nd 2010, 06:30 AMProve It
- May 2nd 2010, 08:36 PMdemode
Prove It,

We know that

If I want to integrate the density function of the normal distribution:

using the value of I, what would be a suitable change of variable in this case? - May 7th 2010, 06:27 PMProve It