The average value of $F\=x y$ over the region shown in Figure 6.1.1(a) is

$\frac{{\int }_0^1{\int }_0^{x\left(1-x\right)}xy\mathit{\ⅆ}y\mathit{\ⅆ}x}{{\int }_0^1{\int }_0^{x\left(1-x\right)}1\mathit{\ⅆ}y\mathit{\ⅆ}x}$ = $\frac{1\/120}{1\/6}$ = $\frac{1}{20}$ 

The numerator is the volume computed in Example 6.2.1, while the denominator is the area computed in Example 6.1.1.

The task template in Table 6.4.1(a) implements, for double integrals (in Cartesian coordinates) that can be iterated in the order $\mathrm{dy} \mathrm{dx}$, the FunctionAverage command in the Student MultivariateCalculus package.

The task template in Table 6.4.1(a) displays the ratio of integrals needed in the computation of the average value. Hence, a solution from first principles entails simply formulating and evaluating these integrals.