Find the volume, on the interval , of a 3-D object whose cross-section at any given point is an equilateral triangle with side length .
In an equilateral triangle, taking one side as the base, the height is the length of the perpendicular bisector to that side. If is the side length in an equilateral triangle, the length of a perpendicular bisector is given by . So the area of a given equilateral triangle in this problem is given by . To find the volume, we integrate the area over the interval. Thus
Thus, the volume of this shape is units.