15.8 - Section 16.8 Triple Integrals in Spherical...

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Unformatted text preview: Section 16.8 Triple Integrals in Spherical Coordinates Integrating Functions in Different Coordinate Systems In the previous section, we used cylindrical coordinates to help evalu- ate triple integrals. In this section we introduce a second coordinate system, called spherical coordinates, to make integrals over spherical regions easier. 1. Spherical Coordinates The second set of coordinates we consider are a little more complicated. They are particularly useful when describing regions or surfaces which are similar to a sphere. Definition 1.1. We define the spherical coordinates ( , , ) of a point P in space as follows: is the distance of P from the origin (so greaterorequalslant 0). is the same angle as in cylindrical coordinates (the angle made from the positive x-axis, so 0 lessorequalslant < 2 ). is the angle between the positive z-axis and the line segment connecting P to the origin (so 0 lessorequalslant lessorequalslant ). phi theta Conversion between these coordinate systems is a little more compli- cated. It is done using the following formulas. Result 1.2. ( i ) The rectangular coordinates of the point ( , , ) in 3 space are x = sin ( )cos ( ), y = sin ( ) sin ( ) and z = cos ( ). ( ii ) The spherical coordinates of the point ( x, y, z ) can be found by solving the equation 2 = x 2 + y 2 + z 2 , and then using the equations given above....
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15.8 - Section 16.8 Triple Integrals in Spherical...

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