Problem

Let $f(x,y)=\dfrac{1}{1+x^2+y^2}$ for $x,y\in(-\infty,\infty)$ and let $D_a=\{(x,y):x^2+y^2\le a^2\}$ for $a>0$. Which of the following statement is true?

(a) $\iint_{D_1} f(x,y)\,d(x,y)=\pi\ln(2)$.

(b) $\iint_{D_2} f(x,y)\,d(x,y)=\pi\ln(4)$.

(c) $\iint_{D_3} f(x,y)\,d(x,y)=\pi\ln(6)$.

(d) $\iint_{D_4} f(x,y)\,d(x,y)=\pi\ln(8)$.

(e) None of the above statements is true.