20080428, 07:23  #12 
"Lucan"
Dec 2006
England
6,451 Posts 

20080429, 01:37  #13 
"Kyle"
Feb 2005
Somewhere near M50..sshh!
2×3×149 Posts 
Alright then, thanks for the help. Because I have never heard of an integral being solevd by symmetry, and my text book does not mention, what are the rules behind it or alternatively, where could I look for more information in a format that is easy enough to understand for a second semester calc student? Thanks.

20080429, 09:53  #14  
Aug 2004
2·5·13 Posts 
Quote:
One obvious rule is that if f is an odd functions, f(x) = f(x), then integrating from a to +a will always give zero, since the integrals from a to 0 and 0 to a cancel out. Similarly, for an even function, f(x) = f(x), integrating from a to a is the same as twice the integral from 0 to a. Remember that mathematicians love to prove things "by symmetry", so it's very often worth looking out for. Chris 

20080501, 06:16  #15 
"Kyle"
Feb 2005
Somewhere near M50..sshh!
1576_{8} Posts 
Thanks Chris...one more thing to point out when integrating an odd function over from a to a...doesn't it have to be a closed interval (i.e. noninfinity) to cancel to zero, otherwise you have to use another technique?

20080501, 06:41  #16  
Aug 2004
202_{8} Posts 
Quote:
Chris 

20080604, 00:56  #17 
Nov 2005
181 Posts 
Note that I'm using the notation that Int(f(x),x,a,b) is the same as the antiderivative of f(x) on the interval from a to b. If a and b are not specified, of course it's an indefinite integral instead.
Well even if at x=0, f(x) is undefined, you can use a limit for something like this: Int(1/x,x,1,1) is unsolvable directly because 1/0 is undefined. Of course since it's an odd function, we already know there's a symmetry, but we'll try to find another solution anyways. If you break it up, you get: (11)(Int(1/x,x,0,1)) but this has a problem since 0/0 is still a hidden possiblity when simplifying this formula. The limit as n goes to z of Int(1/x,x,n,1) is in fact 1/z. Since 0/z=0/0, the final answer is still undefined using this method. Using a table: 0*Int(1/x,x,a,b)=0*(ln(b)ln(a))=0(ln(1)ln(0))=0(ln(0)) Again, you're having trouble since the limit as n> 0 of ln(n) is infinity. Treating the integral as a summation: Another way to look at that though, is that you're subtracting ln(0) from ln(0) and thus the result is zero so long as all the terms cancle. You normally can NOT just add infinite to negative infinity to get zero, but in this case they're the exact SAME infinity so far as the terms are concerned. You'll see this concept when you have to deal with a series assuming you haven't already. A Riemann sum (limit) definition of the integral would have been really handy right about now. ;) http://www.physicsforums.com/archive.../t112179.html http://documents.wolfram.com/v4/MainBook/3.5.8.html http://www.msstate.edu/dept/abelc/math/integrals.html http://en.wikipedia.org/wiki/Antiderivative (especially the discontinous functions section) These links are mainly to help provide more an alternate explanation compared to your textbook. 
20080604, 04:07  #18  
"Richard B. Woods"
Aug 2002
Wisconsin USA
7188_{10} Posts 
Quote:
Symmetry is not limited to explicit geometry. When you consider substituting x for x, you're folding along the yaxis  a basically geometric idea, isn't it? As Chris said, Last fiddled with by cheesehead on 20080604 at 04:08 

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