Question

Find the average value of f(x, y = over the region, r, the triangle with vertices (0, 0, (0, 1 and (1, 1.

Answer 1

Can't be done without knowing what $f$ is...

But I can tell you that the average value of $f$ is given by

$\dfrac{\displaystyle\iint_Rf(x,y)\,\mathrm dx\,\mathrm dy}{\displaystyle\iint_R\mathrm dx\,\mathrm dy}$

At the very least, we can compute the denominator, which is just the area of $R$. You have

$\displaystyle\iint_R\mathrm dx\,\mathrm dy=\int_{x=0}^{x=1}\int_{y=x}^{y=1}\mathrm dy\,\mathrm dx=\int_0^1(1-x)\,\mathrm dx=\dfrac12$

so the average value will be

$2\displaystyle\iint_Rf(x,y)\,\mathrm dx\,\mathrm dy$