Answer
$$\frac{{{a^4}}}{6}$$
Work Step by Step
$$\eqalign{
& \int_0^a {\int_0^{a - x} {\left( {{x^2} + {y^2}} \right)} dy} dx \cr
& {\text{Using a CAS to evaluate the iterated integral}}{\text{.}} \cr
& {\text{From the image we obtain}} \cr
& \int_0^a {\int_0^{a - x} {\left( {{x^2} + {y^2}} \right)} dy} dx = \frac{{{a^4}}}{6} \cr} $$
