Cross Section Example
Situation: You must find the volume of the solid shown below. The base of the solid is the region bounded by the lines f(x) = 1 - x/2,  g(x) = -1 + x/2,  and x = 0. The cross section perpendicular to the x-axis are equilateral triangles.

Cross 1 Cross 2

Solution: The base and area of each triangular cross section are

Length of base
Base = (1 -x/2) - (-1 + x/2) = 2 - x
Area of equilateral triangle
Area = (SQRT(3)/4)(Base)2
Area of cross section
A(x) = (SQRT(3)/4)(2 - x)2.

Because x ranges from 0 to 2, the volume of the solid is
         b                    2
V = §  A(x) dx = §  (SQRT(3)/4)(2 - x)2 dx
         a                    0

                                            2
= -(SQRT(3)/4)[(2 - x)3/3]
                                            0

= (2SQRT(3))/3