Average value function

The ellipse defined by $\mathcal{R}$ is symmetric about all lines that pass through the origin and lie in the same plane as $\mathcal{R}$ . We see that $f$ is a plane in $\mathbb{R^3}$ . If a plane in $\mathbb{R^3}$ is defined as $px+qy$ , then it includes the origin and has an average value of $0$ over any region that is symmetric about all lines that cross the origin in the same plane as that region.

Note that for any function $f = g+h$ , the average value of $f$ over some region $D$ is the sum of the average values of $g$ and $h$ over $D$ . So for some $r \in \mathbb{R}$ , the average value value of $px+qy+r$ over some $D$ with symmetry as described above is the sum of the average values of $px+qy$ and $r$ over $D$ . We already know that the average value of $px+qy$ over $D$ is $0$ . What is left is the constant $r$ , and the average value of $r$ over $D$ is $r$ .

Hence, for $f(x,y) = 7x+13y-9$ , the average value of $f$ over $\mathcal{R} = \boxed{-9}$ .