How to change the order of integration in 3D?

Feb 2016
8
1
Durham
I have been working on a problem that requires me to integrate in the following way:

$$\int_{0}^{\infty} \int_{0}^{y} \int_{0}^{y-x} f(x,y,z)\ dzdxdy$$

I would like to change the order of integration, with dz going last. For the dydxdz option, I've got:

$$\int_{0}^{\infty} \int_{0}^{\infty} \int_{x+z}^{\infty} f(x,y,z)\ dydxdz$$

Is this correct? And what would the dxdydz option look like?

Thanks in advance for any help!
 

SDK

Sep 2016
804
545
USA
I have been working on a problem that requires me to integrate in the following way:

$$\int_{0}^{\infty} \int_{0}^{y} \int_{0}^{y-x} f(x,y,z)\ dzdxdy$$

I would like to change the order of integration, with dz going last. For the dydxdz option, I've got:

$$\int_{0}^{\infty} \int_{0}^{\infty} \int_{x+z}^{\infty} f(x,y,z)\ dydxdz$$

Is this correct? And what would the dxdydz option look like?

Thanks in advance for any help!
Whether you can do this or not depends on $f$. The computations you provided aren't always valid and there is no reason to expect them to be. You have to know ahead of time that both integrals you have written are finite. In that case, then its true they are equal.
 
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