Editing Planimeter (section)

=== Polar coordinates ===
The connection with Green's theorem can be understood in terms of [[Polar coordinate system#Integral calculus (area)|integration in polar coordinates]]: in polar coordinates, area is computed by the integral <math display="inline"> \int_\theta \tfrac{1}{2} (r(\theta))^2\,d\theta,</math> where the form being integrated is ''quadratic'' in ''r,'' meaning that the rate at which area changes with respect to change in angle varies quadratically with the radius.

For a [[parametric equation]] in polar coordinates, where both ''r'' and ''θ'' vary as a function of time, this becomes
<math display="block">\int_t \tfrac{1}{2} (r(t))^2 \, d(\theta(t)) = \int_t \tfrac{1}{2} (r(t))^2\, \dot \theta(t)\,dt.</math>

For a polar planimeter the total rotation of the wheel is proportional to <math display="inline"> \int_t r(t)\, \dot \theta(t)\,dt,</math> as the rotation is proportional to the distance traveled, which at any point in time is proportional to radius and to change in angle, as in the circumference of a circle (<math display="inline"> \int r\,d\theta = 2\pi r</math>).

This last integrand <math display="inline"> r(t) \,\dot \theta(t)</math> can be recognized as the derivative of the earlier integrand <math display="inline"> \tfrac{1}{2} (r(t))^2 \dot \theta(t)</math> (with respect to ''r''), and shows that a polar planimeter computes the area integral in terms of the ''derivative'', which is reflected in Green's theorem, which equates a line integral of a function on a (1-dimensional) contour to the (2-dimensional) integral of the derivative.