johnaspden has it right. To put it more explicitly, we can calculate the area of a circle by integrating the differential element of area dA for an infinitesima

johnaspden has it right. To put it more explicitly, we can calculate the area of a circle by integrating the differential element of area dA for an infinitesimal annulus from 0 to r. Now, dA is simply the arclength (circumference C) times the thickness dr; since the circumference scales linearly with radius, this leads to the integral of a linear function as follows: dA = C dr = τ r dr => A = ½ τ r².