[tex]D[/tex] is a right triangle with base length 1 and height 8, so the area of [tex]D[/tex] is [tex]\dfrac12(1)(8)=4[/tex].
The average value of [tex]f(x,y)[/tex] over [tex]D[/tex] is given by the ratio
[tex]\dfrac{\displaystyle\iint_Df(x,y)\,\mathrm dA}{\displaystyle\iint_D\mathrm dA}[/tex]
The denominator is just the area of [tex]D[/tex], which we already know. The average value is then simplified to
[tex]\displaystyle\frac74\iint_Dxy\,\mathrm dA[/tex]
In the [tex]x,y[/tex]-plane, we can describe the region [tex]D[/tex] as all points [tex](x,y)[/tex] that lie between the lines [tex]y=0[/tex] and [tex]y=8x[/tex] (the lines which coincide with the triangle's base and hypotenuse, respectively), taking [tex]0\le x\le1[/tex]. So, the integral is given by, and evaluates to,
[tex]\displaystyle\frac74\int_{x=0}^{x=1}\int_{y=0}^{y=8x}xy\,\mathrm dy\,\mathrm dx=\frac78\int_{x=0}^{x=1}xy^2\bigg|_{y=0}^{y=8x}\,\mathrm dx[/tex]
[tex]=\displaystyle56\int_{x=0}^{x=1}x^3\,\mathrm dx[/tex]
[tex]=14x^4\bigg|_{x=0}^{x=1}[/tex]
[tex]=14[/tex]
