Question

Given a set \(S \subseteq \mathbb R^2\), how would one find the area of this set? What if it were less-behaved than the standard simple regions?

Answer 1

Would something like \[\iint\limits_{\mathbb R^2} f(x,y) \ dA \text{, if we define } f(x,y) =: \left\{ \begin{array}{ll} 1 & \text{if } (x,y) \in S \\ 0 & \text{if } (x,y) \not\in S \end{array}\right.\] be the best one could do? This doesn't really seem super satisfactory to me, but I guess it would be means of calculating it numerically.

Answer 2

That is one way you could do it.

If S was bounded, then you could look at randomly generating numbers in some rectangle that contains S and calculating the area of S from that

Answer 3

What exactly do you mean by "randomly generating numbers in some rectangle"?

Answer 4

generate random points inside the rectangle.

then the area of S=limits as the number of points selected goes to infinity Area(rectangle)*(number of points that fall into S)/(number of points total)