>>12964255Because the integral of a bounded region gives you the interior of the bounded region, so if you take an integral w.r.t. the radius r, over the circumference (i.e. integrate the value 2 pi r), it returns a value equal to the interior of that region - in this case, the area, pi * r^2.
The derivative is the "inverse" of the integral, so when you take the derivative w.r.t. r of the area pi * r^2, you of course obtain the circumference, 2 pi r.
