Answer
The average value of the linear function $f\left( {x,y} \right) = mx + ny + p$ on the ellipse ${\left( {\frac{x}{a}} \right)^2} + {\left( {\frac{y}{b}} \right)^2} \le 1$ is $p$.
Work Step by Step
The average value of $f\left( {x,y} \right)$ on a domain ${\cal D}$ is given by Eq. (8):
$\bar f = \frac{{\mathop \smallint \nolimits_{}^{} \mathop \smallint \nolimits_{\cal D}^{} f\left( {x,y} \right){\rm{d}}A}}{{\mathop \smallint \nolimits_{}^{} \mathop \smallint \nolimits_{\cal D}^{} 1{\rm{d}}A}}$
So, the average value of the linear function $f\left( {x,y} \right) = mx + ny + p$ on the ellipse ${\left( {\frac{x}{a}} \right)^2} + {\left( {\frac{y}{b}} \right)^2} \le 1$ is
$\bar f = \frac{{\mathop \smallint \nolimits_{}^{} \mathop \smallint \nolimits_{\cal D}^{} \left( {mx + ny + p} \right){\rm{d}}A}}{{\mathop \smallint \nolimits_{}^{} \mathop \smallint \nolimits_{\cal D}^{} {\rm{d}}A}}$
Consider the numerator $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} \left( {mx + ny + p} \right){\rm{d}}A$.
Since the ellipse is symmetric with respect to $x$- and $y$- axis, and if the integrands in the double integrals are odd functions, we conclude that the integrals are zeros. Therefore, we have
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} mx{\rm{d}}A = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} ny{\rm{d}}A = 0$
Thus,
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} \left( {mx + ny + p} \right){\rm{d}}A = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} p{\rm{d}}A$
Since $p$ is a constant, so
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} \left( {mx + ny + p} \right){\rm{d}}A = p\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} {\rm{d}}A$
Therefore,
$\bar f = \frac{{\mathop \smallint \nolimits_{}^{} \mathop \smallint \nolimits_{\cal D}^{} \left( {mx + ny + p} \right){\rm{d}}A}}{{\mathop \smallint \nolimits_{}^{} \mathop \smallint \nolimits_{\cal D}^{} {\rm{d}}A}} = \frac{{p\mathop \smallint \nolimits_{}^{} \mathop \smallint \nolimits_{\cal D}^{} {\rm{d}}A}}{{\mathop \smallint \nolimits_{}^{} \mathop \smallint \nolimits_{\cal D}^{} {\rm{d}}A}} = p$
Hence, the average value of the linear function $f\left( {x,y} \right) = mx + ny + p$ on the ellipse ${\left( {\frac{x}{a}} \right)^2} + {\left( {\frac{y}{b}} \right)^2} \le 1$ is $p$.
