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 Post subject: Integration in R^nPosted: Wed, 7 Mar 2012 19:22:38 UTC
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Here's a problem I am reviewing from analysis. I will list the problem with some commentary below:

Problem: Let be open and have volume, and let be continuous, and for some .
Show that .

Ok, I think saying that A has volume (along with continuity) together forces the integral to exist. I don't think I need to say anything else(?) about this. Now, I remember seeing this back when I did the single variable integral, and I know that continuity means that there exists such that for all .
If my memory is correct, you can then write some inequality, like

I think that is the Calculus II approach. How can I modify this for the setting?

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"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

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 Post subject: Re: Integration in R^nPosted: Wed, 7 Mar 2012 19:41:13 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12102
Location: Austin, TX
Justin wrote:
Here's a problem I am reviewing from analysis. I will list the problem with some commentary below:

Problem: Let be open and have volume, and let be continuous, and for some .
Show that .

Ok, I think saying that A has volume (along with continuity) together forces the integral to exist. I don't think I need to say anything else(?) about this. Now, I remember seeing this back when I did the single variable integral, and I know that continuity means that there exists such that for all .
If my memory is correct, you can then write some inequality, like

I think that is the Calculus II approach. How can I modify this for the setting?

Break up the integration domain into two pieces, one on which for some fixed delta and the other you just care that it's nonnegative. Add the two. In this case you just get where is the size of the ball on which you have a greater than delta value for .

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 Post subject: Re: Integration in R^nPosted: Wed, 7 Mar 2012 20:20:09 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Wed, 21 May 2003 04:27:18 UTC
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Thanks, that definitely works.

_________________
"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

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 Post subject: Re: Integration in R^nPosted: Wed, 7 Mar 2012 22:12:48 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12102
Location: Austin, TX
Justin wrote:
Thanks, that definitely works.

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