Let R be the region bounded by x2 + y2 = 1. Write an integral expression for the volume of the solid with base R
whose slices perpendicular to the x-axis are equilateral triangles.
The region R is the unit circle:
For this solid we slice perpendicular to the x-axis and get equilateral triangles. The triangle at position x has side-length 2y where :
This means the area of the triangle is
The volume of the slice is
and the volume of the entire solid is
Alternately, since the solid is symmetric we could find the area of half of it and then multiply by 2.
In this case, we get the expression