Green's theorem in comparison to normal line integrals

Question

I am having trouble internalizing the geometric significance of Green's Theorem and Line Integrals in general. Why is it that a normal line integral gives area, while Green's Theorem gives volume?

Answer 1

There are three distinct concepts here. First, what you are calling a normal line integral, which I won't go into. Second, the line integral of a vector field. And Third, Green's Theorem.

To be clear, the intuition behind line integrals is not area. This is because, as you point out, this is completely unintuitive. Green's Theorem shoes a relationship between one integral and the other, but a line integral represents a different type of quantity. I could try to explain it, but I honestly think that looking at this gif on Wikipedia might be the most helpful option. If not, I can expand on this.

Third is Green's theorem. The best way to think about Green's theorem is that it's a higher-dimensional analog of the fundamental theorem of Calculus, which states that $\int_a^b F'(x) dx = F(b)-F(a)$. This runs into intuitive problems. After all, the left side is the area under a curve, and the right side is just a sum at two points. However, because we pick the thing in the integral correctly, we can transition between a special one-dimensional integral and a sum (or zero-dimensional integral) along the boundary. Green's theorem does almost exactly the same thing!. It says that if you have a special two-dimensional integral over an area, it's equal to a one dimensional integral over the boundary. In fact, this type of thing is true in any number of dimensions if you rigorously define what it means to be special and what we mean by the boundary.