Can an Integral Be Transformed into a Triple Integral?

[Anna Kaladze]

Dear All,
Sorry perhaps for a silly-looking question from someone who does not have very strong math skills.
In the attached pdf file, I describe a problem which I have been trying to unsuccessfully crack after trying a few manipulations.

Some intuitive thoughts are as follows: the inner two integrals over dy and dd give an area. Perhaps that area depends only on the "height" m. Suppose this area is a sheet of density 1/unit area. The final goal is to integrate E^area over dm.

Is the integral of exponential area of unit density/area = integral of unit area of exponential density/area? Can we pull that exponential "through" the integration?

Any other suggestions helping to transform (1) into a triple integral are highly appreciated.

Thanks a lot.

Anna.
P.S. This is not a h/w question, it is for my own research.

 

[Stephen Tashi]

Have you made any progress on this. Perhaps someone has ideas about a simpler case.


Does [tex]\int_0^t \left[ e^{\int_y^t f(x) dx } \right] dy [/itex]<br /> <br /> capture what your asking about?<br /> <br /> We want to express this as a double integral with the integral signs both to the left of $e$.[/tex]

 

[Anna Kaladze]

Have you made any progress on this. Perhaps someone has ideas about a simpler case.


Does [tex]\int_0^t \left[ e^{\int_y^t f(x) dx } \right] dy [/itex]<br /> <br /> capture what your asking about?<br /> <br /> We want to express this as a double integral with the integral signs both to the left of $e$.[/tex]

$$<br /> Hi Tashi,<br /> Thanks a lor for your reply.<br /> I think it is not possible to trasform that integral the way I want. I had to do a sequence of NIntegrations as oppsed to doing a simple double integral.<br /> Regards,<br /> Anna.$$

 

[chiro]

Dear All,
Sorry perhaps for a silly-looking question from someone who does not have very strong math skills.
In the attached pdf file, I describe a problem which I have been trying to unsuccessfully crack after trying a few manipulations.

Some intuitive thoughts are as follows: the inner two integrals over dy and dd give an area. Perhaps that area depends only on the "height" m. Suppose this area is a sheet of density 1/unit area. The final goal is to integrate E^area over dm.

Is the integral of exponential area of unit density/area = integral of unit area of exponential density/area? Can we pull that exponential "through" the integration?

Any other suggestions helping to transform (1) into a triple integral are highly appreciated.

Thanks a lot.

Anna.
P.S. This is not a h/w question, it is for my own research.

After looking at your integral, (the one inside your exponential), you are going to get a non-trivial region for the double integral that is of course dependent on your function (that is it's not going to be a rectangle or even any static region, but something more complex).

As for turning your equation into a triple integral, good luck with that. I can't think of any transform off the top of my head that will turn your exponential term into a relevant integral. Most transforms I've seen transform standard integrals that a linear into other linear integrals. The fact that you've got this non-linear relationship makes it a lot more complicated.