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 y 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.