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 and 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 and 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 and 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 and the x-axis
between . The angle between the top and bottom of the
wedge is . 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 where 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 .