Re: Problem in Solving Double Integral for PDF transformation

• To: mathgroup at smc.vnet.net
• Subject: [mg80963] Re: Problem in Solving Double Integral for PDF transformation
• From: "Jean-Marc Gulliet" <jeanmarc.gulliet at gmail.com>
• Date: Thu, 6 Sep 2007 05:32:34 -0400 (EDT)
• References: <fblj6o\$pam\$1@smc.vnet.net> <46DE84B3.3010203@gmail.com>

```[See answer at the bottom of the message crossposted to MathGroup.]

Negede Abate wrote:

> Dear Jran-Marc,
>
> Thanks for the help. It seems it might work. But, how can get the
> PiecewiseIntegrate in Mathematica. It is not a built-in function.
>
>
> Negede
>
>
>
> On 9/5/07, Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com> wrote:
> > Negede Abate wrote:
> >
> > > Could any one kindly help me solve the following Integral using
> Mathematica?
> > > I tried the Integrate command  in different forms but I failed to
> succeed. I
> > > really got staked.
> > > *
> > >
> > > f1 = 1/(3*E^((-42.13+t2)^2/18)*Sqrt[2*Pi]);
> > >
> > > f2= (5.778367760333042*
> ^-19*(11.12+t3)^12.13)/E^(4.4008893833458054*^-20*(
> > > 11.12+t3)^13.13);
> > >
> > > f3 = Abs[Cot[0.017453292519943295*t2]];
> > >
> > > f4=0.0037739313253375532/E^(0.000044744314476429635*(
> > > 449.96310600570723-55.69524532558123*t3+1.*t*Cot
> [0.017453292519943295*t
> > > 2])^2);
> > >
> > > f5=
> (7.133612341840616*^-13*(-18.963106005707232+55.69524532558123*t3-1.*t
> > > *Cot[0.017453292519943295*t2])^3.8)/E^(1.4861692378834617*^-13*(-
> > > 18.963106005707232+55.69524532558123*t3-1.*t*Cot
> [0.017453292519943295*t2])^
> > > 4.8);
> > >
> > > Int1=f1 f2 f3 f4 //FullSimplify//InputForm
> > > *
> > >
> > > 2.899933125405506*^-22*
> > >
> > > E^(-0.05555555555555555*(-42.13 + t2)^2 -
> > >
> > > 4.4008893833458054*^-20*(11.12 + t3)^13.13 -
> > >
> > > 0.000044744314476429635*(449.96310600570723 -
> > >
> > > 55.69524532558123*t3 +
> > >
> > > 1.*t*Cot[0.017453292519943295*t2])^2)*
> > >
> > > (11.12 + t3)^12.13*
> > >
> > > Abs[Cot[0.017453292519943295*t2]]
> > > *
> > >
> > > Int2=f1 f2 f3 f5 //FullSimplify//InputForm
> > > *
> > >
> > > 5.481551451404547*^-32*
> > >
> > > E^(-0.05555555555555555* (-42.13 + t2)^2 -
> > >
> > > 4.4008893833458054*^-20*(11.12 + t3)^13.13 -
> > >
> > > 1.4861692378834617*^-13*(-18.963106005707232 +
> > >
> > > 55.69524532558123*t3 -
> > >
> > > 1.*t*Cot[0.017453292519943295*t2 ])^4.8)*
> > >
> > > (11.12 + t3)^12.13*
> > >
> > > Abs[Cot[0.017453292519943295*t2]]*
> > >
> > > (-18.963106005707232 + 55.69524532558123*t3 -
> > >
> > > 1.*t*Cot[0.017453292519943295*t2])^3.8
> > > *
> > >
> > > Integrate[Int1,{t2,35,50},{t3,10,23}]
> > >
> > > Integrate[Int2,{t2,35,50},{t3,10,23}]
> >
> > Glancing at your integrands, I have noticed that they contain piecewise
> > functions such as Abs. In this case, it may be worthwhile to try Maxim
> > Rytin's "Integration of Piecewise Functions with Applications" package
> > available at
> >
> > http://library.wolfram.com/infocenter/MathSource/5117/
> >
> > (Especially, you should check the function *PiecewiseIntegrate*.)
> >
> > From the notebook piecewise.nb, we can read,
> >
> >
> "PiecewiseIntegrate[f,{x,xmin,xmax},{y,ymin,ymax},\[Ellipsis]]
> gives the
> > definite integral of function f. It is intended for integrating
> > piecewise continuous functions, and also generalized functions. It
> > handles integrands and integration bounds involving the following
> > expressions:
> >
> > . UnitStep, Sign, Abs, Min, Max
> > . Floor, Ceiling, Round, IntegerPart, FractionalPart, Quotient, Mod
> > . DiracDelta and its derivatives, DiscreteDelta, KroneckerDelta
> > . If, Which, Element, NotElement
> > . Piecewise, Boole, Clip"

If what you want is just to experiment with PiecewiseExpand during a
Mathematica session, the easiest way should be, first, to open Maxim
Rytin's notebook in Mathematica (if you are using Mathematica version
6, do not scan for possible issues: the core functions work fine with
it; however some of the example might use some deprecated packages),
then evaluate the initialization cells (menu Evaluation -> Evaluate
Initialization Cells in v6 or menu  Kernel -> Evaluate -> Evaluate
Initialization in v5). Now, you can use the function  PiecewiseExpand
from any notebook as long as you are using the same kernel.

If you want to create a package and install it, follow the directions
given at the beginning of the section titled "Implementation" to