Solution
= Solution
For a vertical slice through the <triangle>, $0\leq x\leq2$ and $0\leq y\leq x/2$. Reversing the order by <Fubini's theorem> gives
$$
\int_0^2\left(\int_0^{x/2}x^2y^2\,dy\right)dx=\int_0^2\frac{x^5}{24}\,dx=\boxed{\frac49}.
$$
The agreement checks that both orders describe the same region.