Q8Mathematics I

The original integral defines a solid geometric triangle bounded perfectly by $y=x$, $y=2a-x$, and the vertical axis $x=0$. To reverse the order to $dxdy$, we must slice this specific region horizontally.

Because the right-side bounding curve changes drastically at the vertex $(a,a)$, the region must be mathematically split into a bottom half (bounded strictly by $x=y$) and a top half (bounded strictly by $x=2a-y$). This absolute geometric necessity creates the two separate integrals.