Can Calculus Prove Average Solar Irradiance is a Quarter of Total?

[Dr.Doom]

Homework Statement

I am trying to show that the average solar irradiance over Earth's surface is 1/4 of the total solar irradiance using only calculus.

Homework Equations

Solar Irradiance = S_{o}
Irradiance = F = S_{o}cos(θ)

The Attempt at a Solution

Using geometry, I can easily show that the average solar irradiance is S_{o}/4 by multiplying S_{o} by the ratio of the incident area and total surface area of a sphere:
\frac{S_{o}*∏r^2}{4∏r^2}=\frac{S_{o}}{4}

My question is how can I use calculus to show this? I was thinking that I could integrate S_{o}cos(θ), but I'm not sure what my integration bounds should be. I'm having trouble visualizing how I can integrate over the entire surface area of a sphere.

Any suggestions would be greatly appreciated!

[Edit] I realize this is probably not the right forum to post this question in but I don't know how to change it.

 

[Spinnor]

Googling the definition of solar irradiance your problem came up in one of the links, looks like you need to consider the rotating Earth in your problem.