I wanted to know if there is some general method to identify the upper and lower surfaces for integration while calculating the volume bound by two planes. My particular problem is: Find the volume of the region bounded by $x+z=1$ and $y+2z=2$ in the first octant.
According to the solution we are supposed to perform integration over two different volumes with different surfaces taken as upper and lower, viz.: Draw a line parallel to the $z$-axis and note that the upper surfaces are $2z+y=2$ over the triangle bounded by $x=0,$ $y=1$, $y=2x$, and $z=1-x$ over the triangle bounded by $y=0$, $x=1$, $y=2x$.
Now I know that the general method is to draw a line from $xy$ plane in the positive $z$ direction and identify the surfaces in the order in which the line touches the planes, but this method doesn't work for me, so is there any other way or should I try harder on the visualization part?