Switching order of integration

How do we go from line 1 to line 2?
N is a constant, and x can only take values in [0,1]

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:

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?