You are right. They are not the same The first integral takes x going from 0 to infinity and, for each x, y going from 0 to x. That integral is over the region in the first quadrant bounded by the x-axis, y= 0, and by the line y= x, that is, the area in the first quadrantbelowthe line y= x. The second integral take y going from 0 to infinity and, for each y, x going from 0 to y- the region in the first quadrant bounded by the y-axis, x= 0, and by the line y= x. That is the region in the first quadrantabovethe line y= x.

What would be correct is either

or