average distance interior of disk to center

Question

Find the average distance from a point P(x,y) in the disk x²+y²≤a^2 to the origin.

Answer 1

Assume the origin to be the center of the disk.  We will weigh the distance by the mass: dm = ρdA with dA = 2πrdr, a thin ring around the origin.  Let ρ = M/πa² where M is the mass of the disk.

 

 

r_avg = 1/M ∫rρ(2πr)dr integrated from 0 to a.

 

 

r_avg = (2πρ/M) ∫r² dr = (2πρ/M) [(1/3) r³] from 0 to a.

 

 

r_avg = (2π/M)(M/πa²)(a³/3) = 2a/3