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Re: Double integral of a piecewise-constant function
*To*: mathgroup at smc.vnet.net
*Subject*: [mg61515] Re: Double integral of a piecewise-constant function
*From*: Chris Rodgers <rodgers at physchem.NOSPAMox.aREMOVEc.uk>
*Date*: Fri, 21 Oct 2005 00:38:00 -0400 (EDT)
*Organization*: Oxford University, England
*References*: <200510180644.CAA11181@smc.vnet.net> <dj4q2v$ipv$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
OK. Here is a simpler example where I try to integrate a
piecewise-constant function in two dimensions.
I define a very simple function ("testfunc") with constant values in 1x1
squares over the domain t[1] = 0 to 3 and t[2] = 0 to 3 with value zero
elsewhere.
I then proceed to integrate a triangular region of this surface, whose
integral should be 1+2+3=6. I tried three different approaches:
1) Integrate[Integrate[testfunc, {t[2], 0, t[1]}], {t[1], 0, 3}]
2) Integrate[testfunc, {t[1], 0, 3}, {t[2], 0, t[1]}]
3) Integrate[
Integrate[testfunc, {t[2], 0, t[1]},
Assumptions -> t[1] \[Element] Reals], {t[1], 0, 3}]
In (2) and (3), Mathematica succeeds, but in case (1) it doesn't.
Why does Mathematica not understand that the dummy variable t[1] is Real
in case (1)?
Although this example is trivial, in the work that I am trying to do, it
will be much more difficult to collect all the integrals together into a
single term. Is there any way to make the inner Integrate(s) realise
that t[1] is Real automatically? Can this be scaled up to the case where
I have more than two Integrate's within one-another?
Yours,
Chris Rodgers.
P.S. A workbook containing these formulae and a plot of "testfunc" is
available at
http://physchem.ox.ac.uk/~rodgers/MMA/Problem1.nb
and a PDF showing the output is available at
http://physchem.ox.ac.uk/~rodgers/MMA/Problem1.pdf
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