Polar planimeter
The polar planimeter 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.
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.