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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
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