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Definite integrals in Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg72145] Definite integrals in Mathematica
• From: some guy named Dave <nuclearwhippingboy at hotmail.com>
• Date: Wed, 13 Dec 2006 06:40:29 -0500 (EST)

Hey all,

Hmm, I tried posting this yesterday, but it hasn't shown up yet. So, 
appologies if this gets double-posted.

I'm a newbie to Mathematica, and I have a function that I want to integrate. 
It's basically a smoothing function over a vector space, and I want to 
normalize it. The function itself looks like this:

   f(r) = (h^2 - r^2)^3

where r is the scalar length ||x|| of some displacement vector x, and h is a 
constant. We are really only interested in values of r in [0,h], and for 2D 
vectors. However, I have some research literature involving 3D vectors that I 
want to compare against, just for sanity.

So, if x is a 3D vector, then the surface of the sphere at radius r should be 
(4/3)(f(r))^3.  We should then be able to get the definite integral from 
Mathematica by doing something like:

   Integrate[ 4/3 * Pi( (h^2 - r^2)^3 )^3, {r, 0, h}]

When I do so, I get back

   262144 pi h^19 / 692835

However, the literature I have says that the integral should yield

   64 pi h^9 / 315

So, have I screwed something up? Is the literature wrong, perhaps?

Thanks.

Dave

• Follow-Ups:
  • Re: Definite integrals in Mathematica
    • From: Darren Glosemeyer <darreng@wolfram.com>