Hi. First consider the general expression when integrating with respect to . That's the usual one and is easiest to visualize. It's just integrating from the surface up to the surface between the curves and and going along the x-axis from zero to a. So now that you have
so that means we're integrating between the surfaces to . That's going from the zy-plane to the red plane in the figure. And we're integrating from the lines to which is that green triangle in the zy-plane, and finally we're going from 0 to a in the z direction or just from 0 to 1 in my figure. So it's that little red-purple-green wedge in there that we're integrating over. Now lets switch the order of integration to:
Now we want to integrate first in the z-direction which is from the purple surface up to the black surface so z goes from up to . Now in the y-direction, we go from the line that is to and finally in the x direction we go from 0 to a giving us:
and you can do that integration up to the dx part right? Also, this is a particular case of the general expression:
which I'm pretty sure is derived via differentiation and integration directly.