Re: Problem in Solving Double Integral for PDF transformation

• To: mathgroup at smc.vnet.net
• Subject: [mg80927] Re: Problem in Solving Double Integral for PDF transformation
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Thu, 6 Sep 2007 05:14:08 -0400 (EDT)
• Organization: The Open University, Milton Keynes, UK
• References: <fblj6o\$pam\$1@smc.vnet.net>

```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"

HTH,
--
Jean-Marc

```

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