Calculate Volume of Parabolic Region with Equilateral Triangles

[ewad]

The base of S is an parabolic region {(x,y)|x^2<y<1}. Cross-sections perpendicular to the y-axis are equilateral trinagles
pls help
no clue

 

[slider142]

Draw it out. The cross-section in the yz-plane is half of a parabola: the heights of the triangles.The easiest way to approach the volume of this shape is to use your knowledge of the area of each triangle.

 

[HallsofIvy]

This is a standard homework problem so I am moving this to the homework section.

It will help you to know that if an equilateral triangle has side length s, then it has altitude $\sqrt{3}s$ and so area $(\sqrt{3}/2)s^2$.