Problem integrating what seems like a simple region. I am trying to integrate a region from $0$ to $1$ on the $x$-axis and from $x$ to $1$ on the $y$-axis which is like an upside-down triangle with edges on $y = 1$ and $x = 0$ and $y = x$ so I set up the integral as:
$$\int_0^1 \int_x^1 dy\, dx $$
but when I plug the following code into Wolframalpha:
integrate from 0 to 1 integrate from x to 1 1 dy dx
it gives me the region in the first quadrant under the line $y = -x + 1$ (link to Wolfram Alpha page)

What am I doing wrong?

I suspect the incorrect region is just a problem with Wolfram Alpha. (Note however  that $\frac{1}{2}$ is the correct value of the integral.) The correct region is of course

