Past exam of the mathematical methods II course of the University of Oxford / 2013 resit / 5 / Solution

Write the plane as $z=6-3x-2y$. Its projection on the $xy$ plane is the triangle $0\le x\le 2$, $0\le y\le 3-3x/2$. The scalar surface area element isThus the weighted surface integral isA geometric check uses the triangular face's area $3\sqrt{14}$ and mean $y$ coordinate $(0+3+0)/3=1$. Multiplying them gives the same result, because $y$ is an affine function on this planar triangle. Its edges have zero surface measure, so strict or non-strict octant inequalities do not change the integral.