The polar [url=https://en.wikipedia.org/wiki/Planimeter]planimeter[/url] is composed of two articulated arms. One end O is fixed. The other end S can be moved freely. A graduated wheel, with axis the second arm rubs on the paper. It therefore sums the perpendicular displacements of the arm.[br]By writing the differential form associated with the movement of this wheel, we find the formula at the bottom. The formula at the top is the Cartesian version of the differential 1-form -ydx+xdy whose differential is the 2-form of area dx dy: by integrating this 1-form along a closed loop, we obtain the area of ​​the region bordered by this loop, this is the Green-Riemann theorem.

Move point S slowly along the unit circle to see that numerical integration of the 1-form gives an approximation of the area of ​​the disk. Note that the effective formula is the lower one, which is numerically more unstable than the theoretically equivalent -ydx+xdy form. You can reset the counter at will.