Question

Find the average value of the function σ(x, y) = x + y + x2 + y2 over the disk 0 ≤ x2 + y2 ≤ 4.

Answer 1

The average value of
`\sigma(x,y)` over the disk (call it
`D`) is given by


`(\displaystyle\iint_D\sigma(x,y)\,\mathrm dA)/(\iint_D\mathrm dA)`

The denominator is just the area of
`D`, which we know to be


`\displaystyle\iint_D\mathrm dA=\pi(2)^2=4\pi`

To compute the denominator, convert to polar coordinates, setting


`\begin{cases}x=r\cos\theta\\y=r\sin\theta\end{cases}\implies\mathrm dA=\mathrm dx\,\mathrm dy=r\,\mathrm dr\,\mathrm d\theta`

Then


`\displaystyle\iint_D\sigma(x,y)\,\mathrm dA=\int_(\theta=0)^(\theta=2\pi)\int_(r=0)^(r=2)(r\cos\theta+r\sin\theta+r^2)r\,\mathrm dr\,\mathrm d\theta`

`=\displaystyle\int_(\theta=0)^(2\pi)\left(4+\frac83(\cos\theta+\sin\theta)\right)\,\mathrm d\theta=8\pi`

So the average value of
`\sigma(x,y)` over the disk is
`(8\pi)/(4\pi)=2`.