Find the volume of the solid whose base is the region bounded by the x-axis, the curves y=x, y=3x^2, x=0, and x=.333333 and which has the property that each cross section perpendicular to the x-axis is an equilateral triangle.

Answer 1

You could have just said the region bounded by the two curves, as they intersect at the two given x values.

Each cross-section has a base equal to the distance between the curves, or x-3x^2

Since the area of an equilateral triangle of side s is s^2√3/4, you just have to integrate to add 'em all up to make a volume:

v = ∫[0,1/3] s^2√3/4 dx
where s = x-3x^2
v = ∫[0,1/3] √3/4 (x-3x^2)^2 dx = 1/(1080√3)