Chapter 13.3, Problem 73E

Integrals in strips Consider the integral

     $I={\displaystyle \underset{R}{\iint }\frac{dA}{{\left(1+x^2+y^2\right)}^2}},$

where R = {(x, y): 0 ≤ x ≤ 1, 0 ≤ y ≤ a}.

a.    Evaluate I for a = 1. (Hint: Use polar coordinates.)

b.    Evaluate I for arbitrary a > 0.

c.    Let a → ∞ in part (b) to find I over the infinite strip R = {(x, y): 0 ≤ x ≤ 1, 0 ≤ y ≤ ∞}.