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### Section 6.5 : More Volume Problems

1. Use the method of finding volume from this section to determine the volume of a sphere of radius $$r$$.
2. 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 squares with the base perpendicular to the $$y$$-axis. See figure below to see a sketch of the cross-sections.
3. Find the volume of the solid whose base is a disk of radius $$r$$ and whose cross-sections are rectangles whose height is half the length of the base and whose base is perpendicular to the $$x$$-axis. See figure below to see a sketch of the cross-sections (the positive $$x$$-axis and positive $$y$$-axis are shown in the sketch).
4. Find the volume of the solid whose base is the region bounded by $$y = {x^2} - 1$$ and $$y = 3$$ and whose cross-sections are equilateral triangles with the base perpendicular to the $$y$$-axis. See figure below to see a sketch of the cross-sections.
5. 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 the upper half of the circle centered on the $$y$$-axis. See figure below to see a sketch of the cross-sections.
6. Find the volume of a wedge cut out of a “cylinder” whose base is the region bounded by $$y = \cos \left( x \right)$$ and the $$x$$-axis between $$- \frac{\pi }{2} \le x \le \frac{\pi }{2}$$. The angle between the top and bottom of the wedge is $$\frac{\pi }{4}$$. 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).
7. For a sphere of radius $$r$$ find the volume of the cap which is defined by the angle $$\varphi$$ where $$\varphi$$ is the angle formed by the $$y$$-axis and the line from the origin to the bottom of the cap. See the figure below for an illustration of the angle $$\varphi$$.