Question

Find the average value of f over region

Answer 1

The average value of $f(x,y)$ over the region $\mathcal D$ is

$\dfrac{\displaystyle\iint_{\mathcal D}f(x,y)\,\mathrm dx\,\mathrm dy}{\displaystyle\iint_{\mathcal D}\mathrm dx\,\mathrm dy$

We can describe the region $\mathcal D$ as the set

$\{(x,y)\mid0\le x\le1,\,0\le y\le8x\}$

So the integral in the numerator is

$\displaystyle\iint_{\mathcal D}5xy\,\mathrm dx\,\mathrm dy=5\int_{x=0}^{x=1}\int_{y=0}^{y=8x}xy\,\mathrm dy\,\mathrm dx=40$

The integral in the denominator is the same as the area of $\mathcal D$, which is a right triangle, so it's easy to compute

$\displaystyle\iint_{\mathcal D}\mathrm dx\,\mathrm dy=\frac12\cdot1\cdot8=4$

So the average value is $\dfrac{40}4=10$.