Question On Applying Definite Integrals?

If r(t) is the rate of growth of a certain tree in feet per year where t is in years, what does the expression ∫a=40 b=30 r(t)dt (<– this is supposed to be a definite integral) represent?
I just don't get it… The integral is the area under a curve right? So this is the curve of rate as function of time, and the integral I'm given is the area under it from 30 to 40. What does that represent, quantitatively and qualitatively?
I have a lot of questions that ask me about what exactly the area under a curve represent given a certain function, so I need to understand, fundamentally, what it does so I can work out the other problems by myself.
I dont' have access to my teacher right now so any help is appreciated.