Average Value of a Function notion

[Kreizhn]

Homework Statement

In a textbook that I'm reading, the authors have stipulated that for polarized and isotropic radiation, the angle between the direction of polarization and the electric dipole vector is random, and so we can replace $cos^2 \theta$ by its average 1/3.

I cannot see how this is possible. Can anyone shed some light on this? Am I using the wrong notion of "average"?

Homework Equations

The average value of a function f over an interval [a,b] is given by

$$\displaystyle \frac{1}{b-a} \int_a^b f(x) dx$$

The Attempt at a Solution

I don't believe it is mathematically possible for the average value of cos² to reach 1/3, since if one does the integral, and calculates the absolute minimum value, they should find something along the lines of 0.39, which is greater than 1/3. Thus I must be missing something.

 

[Citan Uzuki]

It's not talking about the average value over an interval, it's talking about the average value over a spherical surface. Specifically, if the direction of the dipole vector is selected randomly from the unit sphere and θ is the angle between it and the polarization vector, then cos² θ will be on average:

$$\frac{1}{4\pi} \int \cos^2 \theta\ \text{d}S$$

Where the integral is taken over the unit sphere. Note that by symmetry, this integral will be the same regardless of what direction the polarization vector is pointing, so picking one and doing the surface integral reveals that this value is indeed 1/3.