Evaluate \(\int\int\int_V (x+y) dV\) where V is the region in the first octant (x ≥ 0, y ≥ 0, z ≥ 0) bounded by the planes x = 1, y = 0, z = 0, and the plane that contains the line y = 2 − x and the point (1, 0, 1).
The equation of the plane containing the line y = 2 - x and the point (1, 0, 1) is
z = 2 - x - y, and this will be the equation of the upper boundary surface, (the top limit for z).
The bottom limit for z is the plane z = 0.
For the area in the xy plane, the upper boundary is the line y = 2 - x, and the lower boundary y = 0.
The limits on x are x = 1 and x = 2.
