Why is the area of a circle of radius r is πr2?

You were right about summing the circumferences of the various circles to arrive at the area of the larger outer circle. However, taking discrete values i. e 1, 2,3 ... will not suffice. Because, you do have circles of radii 0.001, 0.01, 0.1,1 etc till r. Hence, instead of taking a summation from 1 to R-radius of outer circle, you should be looking to calculate the integral between 0 and R.