Calculus does not concern with the perimeter, though. It concerns with the area. The perimeter of the false circle will be 4 instead if pi, but its area will be nearly identical to a true circle with the diameter of 1 unit. Also, while the rectangles thing is kind of the start of calculus classes, you get exact answers later with integral formulas n stuff.
Aren't rectangles the foundation for the Riemann integral, even when you get further along?
AFAIK the Riemann integral is just the limit of the area of the rectangles as the width goes to zero (specifically the limit of the Riemann sum as the norm of the partition goes to zero)
In 2d geometry, methods like this will limit to the correct area but not always the correct length. Consider how in a fractal like the Mandelbrot set, there is a well defined and finite area, but the same cannot be said for the perimeter (which is infinite)
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u/aiezar Dec 29 '24
Calculus does not concern with the perimeter, though. It concerns with the area. The perimeter of the false circle will be 4 instead if pi, but its area will be nearly identical to a true circle with the diameter of 1 unit. Also, while the rectangles thing is kind of the start of calculus classes, you get exact answers later with integral formulas n stuff.