Switching order of integration Watch

DQd
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How do we go from line 1 to line 2?
N is a constant, and x can only take values in [0,1]
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RDKGames
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(Original post by DQd)
How do we go from line 1 to line 2?
N is a constant, and x can only take values in [0,1]
Name:  Capture.PNG
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You have the following integration: \displaystyle \int_{x=0}^{x=1} \int_{v=0}^{v=x}

On an xv-plane, you can sketch this region.






Now consider swapping the order so that you are integrating along x first. For every horizontal strip like in the following diagram, you wish to sum them all up. Clearly, horizontally they begin at x=v and they go all the way up to x=1.




Hence the integration with respect to x will be over the region \displaystyle \int_{x=v}^{x=1}.

Then integrating with respect to v, well it should be obvious that we only need the part that is between v=0 and v=1 hence the order of integration becomes

\displaystyle \int_{v=0}^{v=1} \int_{x=v}^{x=1}
Last edited by RDKGames; 3 months ago
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DQd
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(Original post by RDKGames)
You have the following integration: \displaystyle \int_{x=0}^{x=1} \int_{v=0}^{v=x}

On an xv-plane, you can sketch this region.






Now consider swapping the order so that you are integrating along x first. For every horizontal strip like in the following diagram, you wish to sum them all up. Clearly, horizontally they begin at x=v and they go all the way up to x=1.




Hence the integration with respect to x will be over the region \displaystyle \int_{x=v}^{x=1}.

Then integrating with respect to v, well it should be obvious that we only need the part that is between v=0 and v=1 hence the order of integration becomes

\displaystyle \int_{v=0}^{v=1} \int_{x=v}^{x=1}
Thanks, it helps a lot to see these things visually.
Am I correct in saying the "outer" integral prevents the following from happening:
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I guess you can do this visually for any double integral with 2 variables & moving limits? just remembering that 'strips' going from L to R and bottom to up are positive ""areas"", and the opposite way negatives?
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RDKGames
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(Original post by DQd)
Thanks, it helps a lot to see these things visually.
Am I correct in saying the "outer" integral prevents the following from happening:
In integral forms we are saying 'integrate from this line to this line' but it's not enough to trap a closed region because we could be integrating forever, as shown by your images. Which is precisely what the second integration is for.

I guess you can do this visually for any double integral with 2 variables & moving limits? just remembering that 'strips' going from L to R and bottom to up are positive ""areas"", and the opposite way negatives?
Yes sketching these areas of integration is very useful when swapping order of integration.

It is slightly less obvious for a triple integral (or higher orders).
Last edited by RDKGames; 3 months ago
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