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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