Question

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 helpful at all and I'm starting to fail tests, please help

Answer 1

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...