I am supposed to integrate the function $f(x,y)= \exp(-y^2)$over the region $[0,1]\times[x,1]$.
I know I can take the double integral for each $x$ and $y$ from Fubini’s theorem, but I am stuck with the error function after doing so and I strongly believe that I am missing something.
Hint: Using Fubini, you can perform the integration as an iterated integral: $$ \int_{x=0}^1\left(\int_{y=x}^1 \exp(-y^2)\,dy\right)\,dx\tag1 $$ Written this way, where you integrate over $y$ first, leads to the error function. Instead, swap the order of integration (which is permitted by Fubini), so that you are integrating over $x$ first. To see what the limits of integration should be, write $f(x,y)$ as $$ f(x,y)= \exp(-y^2)I_A(x,y)$$ where $A$ is the region in the plane $\{(x,y): 0\le x\le1\ \text{and}\ x\le y\le 1\}$ , so $A$ is the triangle with vertices $(0,0)$, $(1,1)$, $(0,1)$.