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]

capture what your asking about?

We want to express this as a double integral with the integral signs both to the left of $e$.

 

[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]

capture what your asking about?

We want to express this as a double integral with the integral signs both to the left of $e$.

Hi Tashi,
Thanks a lor for your reply.
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.
Regards,
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.

 

[CellCoree]

Dear Anna,

Thank you for sharing your problem with us. It seems like you have made some progress in trying to solve the integral, but are now stuck on how to transform it into a triple integral. Here are some suggestions that may help you in your efforts:

1. Use the properties of integrals: Remember that integrals have certain properties that can help you manipulate them. One such property is the linearity property, which states that the integral of a sum is equal to the sum of the integrals. So, you can try splitting the integral into smaller parts and then combining them back together.

2. Change the order of integration: Sometimes, changing the order of integration can make the integral easier to solve. In your case, you have a double integral over dy and dd, so you can try changing the order of integration to dd and dy instead. This may simplify the integral and make it easier to transform into a triple integral.

3. Use substitution: If you are able to identify a substitution that can simplify the integral, you can use it to transform the integral into a simpler form. This may also help in transforming it into a triple integral.

Regarding your question about the exponential, the integral of an exponential function is not necessarily equal to the exponential of the integral. However, if you are able to transform the integral into a triple integral, you may be able to use the properties of integrals to simplify it further and potentially apply the exponential function to it.

I hope these suggestions are helpful to you in solving your problem. Keep exploring and don't give up, as sometimes the solution can come from trying different approaches. Good luck with your research!

Best regards,