How do I get the area of a circle using double integrals?

The region bounded by a circle is given by the set

R = { (x,y) : -r ≤ x ≤ r, -√(r² - x²) ≤ y ≤ √(r² - x²) }.

As a double integral, the area is

A = ∫ {x = -r to r} ∫ {y = -√(r² - x²) to √(r² - x²)} dy dx

As it is, it's a very difficult integral to compute. Changing to polar coordinates simplifies things. The same region is given by the set (in polar coordinates)

R = { (t,Θ) : 0 ≤ t ≤ r, 0 ≤ Θ ≤ 2π }

As an integral,

A = ∫ {t = 0 to r} ∫ {Θ = 0 to 2π} t dt dΘ

(Note: polar coordinates are conventionally denoted (r, Θ), but I use a t so as to avoid confusion with the constant radius r of the circle.)