Integrals as Probabilities Firstly, I'm not a mathematician as will become evident in a quick moment. I was pondering some maths the other day and had an interesting thought: If you encased an integrable function over some range in a primitive with an easily computable area, the probability that a random point within said primitive also exists below that function's curve, scaled by the area of the primitive, is the indefinite integral of the function over that domain.
So let's say I want to "solve" for $\pi$. Exploiting a circle's symmetry, I can define $\pi$ as:  
$$4 \int_{0}^{1}\sqrt{1-x^2} \,dx$$
Which I can "encase" in the unit square. Since the area of the unit square is 1, $\pi$ is just 4 * the probability that a point chosen at random within the unit square is below the quarter-circle's arc.
I'm sure this is well known, and so my questions are:


*

*What is this called?

*Is there anything significant about
this--for instance, is the
relationship between the integral and
the encasing object of interest--or
is it just another way of phrasing
indefinite integrals?
Sorry if this is painfully elementary!

This is known as geometric probability.
