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 Post subject: very difficult definite integralPosted: Mon, 9 Apr 2012 16:31:57 UTC
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Joined: Thu, 15 Dec 2011 16:54:36 UTC
Posts: 16
The following integral can be used to calculate the moment of intertia of the nucleus of an atom. How do I go about solving the below Moment of inertia integral?
(It is very complicated because of a variable density)

I= ρ* ∫∫∫ r^4*sin^3(θ)* [1+e^((r-R)/a)]^(-1) dr dθ dφ

dr is integrated from 0 to R

dθ is integrated from 0 to pi

dφ is integrated from 0 to 2pi

Any advice, or hints would be greatly appreciated. Are there number calculators that could solve this?

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 Post subject: Re: very difficult definite integralPosted: Mon, 9 Apr 2012 16:51:43 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
jjbyram wrote:
The following integral can be used to calculate the moment of intertia of the nucleus of an atom. How do I go about solving the below Moment of inertia integral?
(It is very complicated because of a variable density)

I= ρ* ∫∫∫ r^4*sin^3(θ)* [1+e^((r-R)/a)]^(-1) dr dθ dφ

dr is integrated from 0 to R

dθ is integrated from 0 to pi

dφ is integrated from 0 to 2pi

Any advice, or hints would be greatly appreciated. Are there number calculators that could solve this?

The - and -integrals are easy. The -integral is not expressible in closed form using elementary functions alone (you can play with polylog if you want...).

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 Post subject: Re: very difficult definite integralPosted: Mon, 16 Apr 2012 19:05:41 UTC
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Joined: Thu, 15 Dec 2011 16:54:36 UTC
Posts: 16
Supposed the integral was simplified:

I= ρ* ∫∫∫ r^4*sin^3(θ)* [1+e^(r/a)]^(-1) dr dθ dφ

dr is integrated from 0 to R

dθ is integrated from 0 to pi

dφ is integrated from 0 to 2pi

Can this now be solved with integration by parts?

I saw that on a lookup table:

∫ [a+be^(px)]^(-1) dx = x/a - (1/ap)*log(a+be^px)

if so, how would the integration of parts be set up?

Most appreciatively,

Jeff

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 Post subject: A nasty u-substitution integral?Posted: Tue, 17 Apr 2012 20:18:00 UTC
 Member

Joined: Thu, 15 Dec 2011 16:54:36 UTC
Posts: 16
I am tryining to solve the following integral for I, the moment of inertia ( ρ and a are constants):

I= ρ* ∫∫∫ r^4*sin^3(θ)* [1+e^(r/a)]^(-1) dr dθ dφ

dr is integrated from 0 to R

dθ is integrated from 0 to pi

dφ is integrated from 0 to 2pi

Can this now be solved with integration by parts?

I saw that on a lookup table:

∫ [ [a+be^(px)]^(-1) ] dx = x/a - (1/ap)*log(a+be^px)

if so, how would the integration of parts be set up?

Most appreciatively,

Jeff

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 Post subject: Re: A nasty u-substitution integral?Posted: Tue, 17 Apr 2012 21:50:02 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12103
Location: Austin, TX
jjbyram wrote:
I am tryining to solve the following integral for I, the moment of inertia ( ρ and a are constants):

I= ρ* ∫∫∫ r^4*sin^3(θ)* [1+e^(r/a)]^(-1) dr dθ dφ

dr is integrated from 0 to R

dθ is integrated from 0 to pi

dφ is integrated from 0 to 2pi

Can this now be solved with integration by parts?

I saw that on a lookup table:

∫ [ [a+be^(px)]^(-1) ] dx = x/a - (1/ap)*log(a+be^px)

if so, how would the integration of parts be set up?

Most appreciatively,

Jeff

First use the constant limits to switch the order and do the integral first, replace a with and then let so that , then do the integral next since there's no dependence, that comes out immediately. For the last part it looks like maybe is called for.

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 Post subject: Re: very difficult definite integralPosted: Wed, 18 Apr 2012 04:56:21 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Topic merged.

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