Final answer:
The average distance from the origin in the unit disk centered at the origin is 1/3 units.
Step-by-step explanation:
To find the average distance from the origin in the unit disk centered at the origin, we need to calculate the average radial position of all points within the disk.
For a unit disk centered at the origin, we can use polar coordinates to express the position of any point as (r, θ), where r is the radial distance from the origin and θ is the angle with the positive x-axis.
Since the disk is symmetric about the origin, the average radial position will be the same in all directions. Therefore, we can calculate it by finding the average value of r over the entire disk.
The average value of r for a unit disk can be found using the integral concept as follows:
∞ ∫ r * (2πr) dr = 0 to 1 ∫ r^2 dr = 1/3
Therefore, the average distance from the origin in the unit disk is 1/3 units.