Spherical coordinates make it simple to describe a sphere just as cylindrical coordinates make it easy to describe a cylinder. If the region has a more natural expression in polar coordinates or if f has a simpler antiderivative in polar coordinates then the change in polar coordinates is appropriate.
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As we have seen earlier in two-dimensional space a point with rectangular coordinates can be identified with in polar coordinates and vice versa where and are the relationships between the variables.
. In this case the triple describes one distance and two angles. So given a polar plot partition the interval into equally spaced subintervals as. The length of each subinterval is representing a small change in angle.
In three-dimensional space a point with rectangular coordinates can be identified with cylindrical coordinates and vice versa. Grid lines for spherical coordinates are based on angle. Complexity of integration depends on the function and also on the region over which we need to perform the integration.
Review of Cylindrical Coordinates. The area of the region defined by the th subinterval can be approximated with a sector of a circle with radius for some in. Otherwise use rectangular coordinates.
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