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andreyandreev [35.5K]
3 years ago
5

Find the average value of f over region

Mathematics
1 answer:
Sedaia [141]3 years ago
7 0

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.

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