The exact area is PI.
I posted this question in anticipation of today, Nerd Appreciation Day.
If I remember correctly the techniques used to calculate the area are from Calculus II.
The anti-derivative of f(x) = 1/(1 + x^2) is F(x) = tan^-1 (x).
Evaluated at the limit of x as x gets closer to infinity is (PI/2), evaluated at x as x gets closer to negative infinity is (-PI/2), (PI/2) – (-PI/2) = PI.
QED