# 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?

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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
In this case it doesn't matter since you are integrating half of a square. More precisely, the upper half of the square with a line going cutting it in half diagonally. You know that for a square, $Area=4*length.$ So, just take half of that and you get your integral which is equal to $\frac12$.