My question is a bit vague, but I'm trying to get a better understanding of surface integrals and their relation to physics. Suppose I have a surface, say a sphere, and I have a function which gives the temperature of the sphere at any point. My textbook says that if I want to find the average temperature of a point on the surface of the sphere, I integrate this temperature function over the entire surface and divide by the surface area. I can understand this intuitively - the integral is like the infinitesimal sum of area times temperature, so dividing through by area gives the average temperature. What I don't understand is what physical quantity the value of the integral (before division) represents? I know it will have units temperature*m^2 but that's it.
Thanks for any replies.
Since thermal energy $E = c T$ is proportional to temperature (in Kelvin), then the integral will represent (up to a multiplicative constant) the total thermal energy of the sphere. The constant is not non-dimensional, though, and depends on the properties of the material.