Find the volume of the solid whose base is the region in the first quadrant bounded by y=x^3, y=1, and the y axis whose cross sections perpendicular to the y axis are equilateral triangles.

The region in question is the set

or equivalently,

Cross sections are taken perpendicular to the y-axis, which means each section has a base length equal to the horizontal distance between the curve y = x³ and the line x = 0 (the y-axis). This horizontal distance is given by

y = x³   ⇒   x = ∛y

so that each triangular cross section has side length ∛y.

The area of an equilateral triangle with side length s is √3/4 s², so each cross section contributes an infinitesimal area of √3/4 ∛(y²).

Then the volume of this solid is

I've attached some sketches of the solid with 16 and 64 such cross sections to give an idea of what this solid looks like.

29 mil

step-by-step explanation: