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### Section 6-5 : More Volume Problems

1. Find the volume of a pyramid of height $$h$$ whose base is an equilateral triangle of length $$L$$. Solution
2. Find the volume of the solid whose base is a disk of radius $$r$$ and whose cross-sections are squares. See figure below to see a sketch of the cross-sections. Solution
3. Find the volume of the solid whose base is the region bounded by $$x = 2 - {y^2}$$ and $$x = {y^2} - 2$$ and whose cross-sections are isosceles triangles with the base perpendicular to the $$y$$-axis and the angle between the base and the two sides of equal length is $$\frac{\pi }{4}$$. See figure below to see a sketch of the cross-sections. Solution
4. Find the volume of a wedge cut out of a “cylinder” whose base is the region bounded by $$y = \sqrt {4 - x}$$, $$x = - 4$$ and the $$x$$-axis. The angle between the top and bottom of the wedge is $$\frac{\pi }{3}$$. See the figure below for a sketch of the “cylinder” and the wedge (the positive $$x$$-axis and positive $$y$$-axis are shown in the sketch – they are just in a different orientation). Solution