I'm reading through the Wikipedia article on Jordan measure and I understand how one could use it to find the area or volume of a 2- or 3-dimensional region. But how would one find the length of a plane curve or space curve?
In particular, the article says that you find the inner Jordan measure by taking the infimum of the collections of disjoint rectangles strictly contained in the region and similarly for the outer Jordan measure the supremum of the collections of disjoint rectangles that strictly contain the region. Then your set is Jordan measureable if those two values are equal. But a simple curve like a circular arc would neither be contained in any line segment (a 1-rectangle) nor contain any nondegenerate line segments.
So how can we use Jordan measure to find the length of a curve? I'm fine with the regular calculus idea of approximating a curve by a bunch of line segments as in the image
but that doesn't seem to be the idea with Jordan measure (though I could certainly be misunderstanding). Are "curved" sets simply not Jordan measureable?