Greens Theorem & does the perimeter of a double integral over a region converge to the perimeter of the original region?

Question

When I've seen double integrals presented, usually its visualized as adding a bunch of small rectangular dA elements along the region. It feels pretty reasonable that this converges to the area.

What about the perimeter of the region? When we make these little squares to approximate the region, would the outer perimeter of the outer squares converge to the perimeter of the original region?

It feels uncomfortable to say this is the case as the staircase paradox approaches the area but not the perimeter. A sketch of a proof for Green's Theorem reflects this. Is there any way to convince myself that the perimeter converges (or the line integral) converges when we sum all these small rectangles?

Probably not, as this would be an example of the staircase paradox or the related $\pi=4$ — Ninad Munshi, Jan 18 '21 at 07:56
I had those examples in mind which made me feel unsure. However, I ask this due to a sketch of a proof of Green's theorem, where it feels like the perimeters should converge for it to be true greens theorem sketch — JCK, Jan 18 '21 at 17:56
So it could be true, but it's also possible that it's saying different things. Perhaps the line integrals converge even if the perimeters don't. — Ninad Munshi, Jan 18 '21 at 18:10
Oh great point, I will update my question to reflect that. To me it seemed if the perimeter converges then the line integral converges — JCK, Jan 18 '21 at 18:57

Greens Theorem & does the perimeter of a double integral over a region converge to the perimeter of the original region?

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