As a circle expands, its area grows proportional to its circumference. The circumference is proportional to the radius.
So you're getting the area by integrating a linear thing.
So you're getting the area by integrating a linear thing.
Exactly. In symbols, this reads dA/dr = C ∝ r. Setting τ = C/r = 6.2831853…, we have dA = C dr = τ r dr => A = ½ τ r². Q.E.D.