Solution
= Solution
The region is the <triangle> $0\leq y\leq x\leq1$. Reversing the integration order makes the inner <integral> elementary:
$$
\int_0^1\int_y^1e^{x^2}\,dx\,dy
=\int_0^1\int_0^xe^{x^2}\,dy\,dx
=\int_0^1xe^{x^2}\,dx
=\boxed{\frac{e-1}{2}}.
$$
The integrand is continuous on this compact <triangle>, so <Fubini theorem> justifies exchanging the order.