Note what you are trying to do here. You are finding the volume of the solid via the method of slicing:

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

So,

Thus,

Can you try to make sense out of what I did?

--Chris