Note that, in spherical coordinates
Consequently, in spherical coordinates, the equation of the sphere is and the equation of the cone is Let’s write with Here is a sketch of the part of the ice cream cone in the first octant. The volume of the full ice cream cone will be four times the volume of the part in the first octant.
We shall cut the first octant part of the ice cream cone into tiny pieces using spherical coordinates. That is, we shall cut it up using planes of constant cones of constant and spheres of constant
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First slice the (the first octant part of the) ice cream cone into segments by inserting many planes of constant with the various values of differing by The figure on the left below shows one segment outlined in red. Each segment
has essentially constant on the segment, and
has running from to and running from to
The leftmost segment has, essentially, and the rightmost segment has, essentially, See the figure on the right below.
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Concentrate on any one segment. A side view of the segment is sketched in the figure on the left below. Subdivide it into long thin searchlights by inserting many cones of constant with the various values of differing by The figure on the left below shows one searchlight outlined in blue. Each searchlight
has and essentially constant on the searchlight, and
has running over
The leftmost searchlight has, essentially, and the rightmost searchlight has, essentially, See the figure on the right below.
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Concentrate on any one searchlight. Subdivide it into tiny approximate cubes by inserting many spheres of constant with the various values of differing by The figure on the left below shows the side view of one approximate cube in black. Each cube
has and all essentially constant on the cube and
The first cube has, essentially, and the last cube has, essentially, See the figure on the right below.
Now we can build up the volume.
Concentrate on one approximate cube. Let’s say that it contains the point with spherical coordinates
The cube has volume essentially
by
3.7.3.
To get the volume any one searchlight, say the searchlight whose
coordinate runs from
to
we just add up the volumes of the approximate cubes in that searchlight, by integrating
from its smallest value on the searchlight, namely
to its largest value on the searchlight, namely
The volume of the searchlight is thus
To get the volume of any one segment, say the segment whose
coordinate runs from
to
we just add up the volumes of the searchlights in that segment, by integrating
from its smallest value on the segment, namely
to its largest value on the segment, namely
The volume of the segment is thus
To get the volume of the part of the ice cream cone in the first octant, we just add up the volumes of the segments that it contains, by integrating from its smallest value in the octant, namely to its largest value on the octant, namely
The volume in the first octant is thus
So the volume of
the total (four octant) ice cream cone, is
We can express (which was not given in the statement of the original problem) in terms of (which was in the statement of the original problem), just by looking at the triangle
The right hand and bottom sides of the triangle have been chosen so that which was the definition of So and the volume of the ice cream cone is
Note that, as in Example
3.2.11, we can easily apply a couple of sanity checks to our answer.
If so that the cone is just which is the line the total volume should be zero. Our answer does indeed give in this case.
In the limit the angle and the ice cream cone opens up into a hemisphere of radius Our answer does indeed give the volume of the hemisphere, which is