The base of a solid in the xy-plane is the first-quadrant region bounded y = x and y = x^2. cross sections of the solid perpendicular to the x-axis are equilateral triangles. what is the volume, in cubic units, of the solid?

my teacher isn't at all and i'm starting to fail tests,

He area of an equilateral triangle of side "s" is s^2*sqrt(3)/4. So the volume of the slices in your problem is 

(x - x^2)^2 * sqrt(3)/4. 
Integrating from x = 0 to x = 1, we have 
[(1/3)x^3 - (1/2)x^4 + (1/5)x^5]*sqrt(3)/4 
= (1/30)*sqrt(3)/4 = sqrt(3)/120 = about 0.0144. 

Since this seems quite small, it makes sense to ask what the base area might be...integral from 0 to 1 of (x - x^2) dx = (1/2) - (1/3) = 1/6. Yes, OK, the max height of the triangles occurs where x - x^2 = 1/4, and most of the triangles are quite a bit shorter...