HOW DO YOU SEE IT? The solid is bounded below by the upper nappe of a cone and above by a sphere (see figure). Would it be easier to use cylindrical coordinates or spherical coordinates to find the volume of the solid? Explain.
To Calculate: Whether it would be easier to use cylindrical coordinates or spherical coordinates to calculate volume of the given solid bounded below by the upper nappe of a cone and above by a sphere as shown in figure in the question.
Refer to figure in question.
The solid is bounded below by the upper nappe of a cone and above by a sphere .
Volume of a solid in cylindrical coordinates is,
And volume of a solid in spherical coordinates is,
It is clear from the figure that varies from (from upper nape of the cone) to (surface of the sphere).
Consider the general equation of sphere that is
Where is the radius of sphere. Therefore, radius of the provided sphere is,
Now the cone and sphere intersects at,
Take square root on both the sides,
It is known that in spherical coordinate .
Where is the angle made by plane or to make sphere .
Substitute the value of and in equation to find the value of ,
Thus, bounds of is,
For sphere angle varies from
Hence, volume of provided solid is,
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