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


`(\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=\frac12`

so the average value will be


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