First plot the triangular region. Once you do that, you can find the equation of the line that is created, connecting the points (0,1) and (1,0). I leave it for you to show that the line is .
Now, what do we do about each individual cross section?
Recall that the area of a triangle is
The base here, is the distance from our line to the x axis:
I leave it for you to show that the height of the equilateral triangle is
Can you try to make sense out of what I did?