Solution:

$\begin{aligned} \text { let } I=& \int_{0}^{a} \int_{\sqrt{a^{2}-x^{2}}}^{x+3 a} f(x, y) d x \ d y \\ \text { Region of integration is: } & \sqrt{a^{2}-x^{2}} \leq y \leq x+3 a \\ 0 & \leq x \leq a \end{aligned}$

Intersection of $x=a$ and $y=x+3 a$ is $(a, 4 a)$

Intersection of $x=0$ and $y=x+3 a$ is $(0,3 a)$

Divide the region into three parts $R 1, R 2$ and $R 3$

$\therefore R=R 1 \cup R 2 \cup R 3$

$\begin{aligned} \text { For region R1: } & \sqrt{a^{2}-y^{2}} \leq x \leq a \\ 0 & \leq y \leq a \\ \text { For region } R 2 : & 0 \leq x \leq a \\ & a \leq y \leq 3 a \\ \text { For region R3: } &(y-3 a) \leq x \leq a \end{aligned}$

$3 a \leq y \leq 4 a$

After changing the order of integration fro dy dx to dx dy

$\therefore I=\int_{0}^{a} \int_{\sqrt{a^{2}-y^{2}}}^{a} f(x, y) d x \ d y+\int_{a}^{3 a} \int_{0}^{a} f(x, y) d x \ d y+\int_{3 a}^{4 a} \int_{(y-3 a)}^{4 a} f(x, y) d x \ d y$