The average value of [tex]f(x,y)[/tex] over the region [tex]\mathcal D[/tex] is
[tex]\dfrac{\displaystyle\iint_{\mathcal D}f(x,y)\,\mathrm dx\,\mathrm dy}{\displaystyle\iint_{\mathcal D}\mathrm dx\,\mathrm dy[/tex]
We can describe the region [tex]\mathcal D[/tex] as the set
[tex]\{(x,y)\mid0\le x\le1,\,0\le y\le8x\}[/tex]
So the integral in the numerator is
[tex]\displaystyle\iint_{\mathcal D}5xy\,\mathrm dx\,\mathrm dy=5\int_{x=0}^{x=1}\int_{y=0}^{y=8x}xy\,\mathrm dy\,\mathrm dx=40[/tex]
The integral in the denominator is the same as the area of [tex]\mathcal D[/tex], which is a right triangle, so it's easy to compute
[tex]\displaystyle\iint_{\mathcal D}\mathrm dx\,\mathrm dy=\frac12\cdot1\cdot8=4[/tex]
So the average value is [tex]\dfrac{40}4=10[/tex].
