I'm studying calculus and I'm having some basic questions, this one is regarding the area of a circle. we know, from some guy, that the circumference of a circle is $2 \pi r$ and the area can be seen as the sum of all the circumferences from $0$ to $r$ which is the same as the integral, which leads us to $\pi r^2$. 
my question is how is it that $$\int\_0^R 2\pi r\ dr = \text{Area}\ \ ?$$