Answer
The average distance from points in $D $ to the origin is given by:
$$
\begin{aligned}
f_{\text {ave }} &=\frac{1}{A(D)} \iint_{D} \sqrt{x^{2}+y^{2}} d A \\
&=\frac{2}{3} a.
\end{aligned}
$$
Work Step by Step
Let $D$ be the disk with center at the origin and radius $a$.
The region $D$ can be described as:
$$ D=\left\{(r, \theta) | 0 \leq r \leq a, \quad 0 \leq \theta \leq 2\pi \right\} $$
the distance from the point $(x,y)$ to the origin is the function $f(x,y)=\sqrt {x^{2}+y^{2}}$, so the average distance from points in $D $ to the origin is
$$
f_{\mathrm{ave}}=\frac{1}{A(D)} \iint_{R} f(x, y) d A
$$
where $A(D)=\pi a^{2}$. So, the average value of the given function is given by:
$$
\begin{aligned}
f_{\text {ave }} &=\frac{1}{A(D)} \iint_{D} \sqrt{x^{2}+y^{2}} d A \\
& =\frac{1}{\pi a^{2}} \int_{0}^{2 \pi} \int_{0}^{a} \sqrt{r^{2}} r d r d \theta \\
&=\frac{1}{\pi a^{2}} \int_{0}^{2 \pi} d \theta \int_{0}^{a} r^{2} d r \\
&=\frac{1}{\pi a^{2}}[\theta]_{0}^{2 \pi}\left[\frac{1}{3} r^{3}\right]_{0}^{a} \\
&=\frac{1}{\pi a^{2}} \cdot 2 \pi \cdot \frac{1}{3} a^{3} \\
&=\frac{2}{3} a.
\end{aligned}
$$
