Double integral of $\exp(-t^2)$ on a triangular domain My calculus teacher gave us this interesting problem: Calculate
$$ \int_{0}^{1}F(x)\,dx,\ $$ where $$F(x) = \int_{1}^{x}e^{-t^2}\,dt $$
The only thing I can think of is using the Taylor series for $e^{-t^2}$ and go from there, but since we've never talked about uniform convergence and term by term integration, I suppose that there is an easier way to do this.

Hint: First (1) flip the limits on the inner integral (assuming those limits are correctly stated), then (2) switch the order of integration:
$$
\int_{x=0}^1\int_{t=1}^x e^{-t^2}dt\,dx
\stackrel{(1)}=-\int_{x=0}^1\int_{t=x}^1 e^{-t^2}dt\,dx
\stackrel{(2)}=-\int_{t=0}^1\int_{x=0}^t e^{-t^2}dx\,dt
=-\int_{t=0}^1t e^{-t^2}dt
$$
