Example 2: Cylindrical Wedge

A wedge is cut from a cylindrical tree trunk of radius 3 ft, by cutting the tree with two planes meeting on a line through the axis of the cylinder. The wedge is 1 ft thick at its thickest point. Find its volume.

Figure 6.1.8 Example 2

The wedge is shown in Figure 6.1.8. The cross sections perpendicular to the x-axis are similar triangles. Place the edge along the x-axis with x from - 3 to 3. At the thickest point, where x = 0, the cross section is a triangle with base 3 ft and altitude 1 ft. The base of the cross section triangle at x is

b =

and the altitude is

⅓b = ⅓

The area of the cross section is

A(x) = ½ · base · altitude = ½b · ⅓b = 1/6b² = 1/6(9 - x²).

The volume is thus

The solution is 6 cubic feet.