How the Roll of the Wheel is Related to Area -- The Full Story

Of course the full truth is more complicated, but only slightly more.  If the motion of the planimeter is very small, then it is more plausible that the small oriented area swept out is the sum of the areas due to a small translation and a small rotation. We have

dA  =  l ds  -  1/2 l l 2 dq,

where dA, ds, and dq denote the small area, the small amount the wheel rolls, and the small angle of rotation of the planimeter.  If we consider the full motion of the planimeter as being broken up into very small movements, then the formula above holds for each of the small movements.  Adding them up yields

A  =  l s  -  1/2 l l 2 Dq

for the total oriented area swept out.

Calculus is used to make this rigorous. Calculus tells us that the first formula above holds at the infinitesimal level, since the infinitesimal change of a dependent variable due to the infinitesimal changes of two independent variables is the sum of the changes caused by each independent variable separately.1  The oriented area made by a large motion of the planimeter is obtained by integrating the first formula above, yielding the second formula.  This integration is easy because the coefficients are constant.

1Here it is in symbols. Suppose z = f(x, y).  If x changes infinitesimally by dx and y stays fixed, the infinitesimal change of z is fx dx, where fx denotes the rate of change (or partial derivative) of f with respect to x.  Similarly, if y changes infinitesimally by dy and x stays fixed, the infinitesimal change of z is fy dy.  The infinitesimal change of z caused by infinitesimal changes of both x and y is dz = fx dx + fy dy.