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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 ( ̺, ϑ, ϕ ) of a point P in space as follows: • ̺ is the distance of P from the origin (so ̺ greaterorequalslant 0). • ϑ is the same angle as in cylindrical coordinates (the angle made from the positive xaxis, so 0 lessorequalslant ϑ < 2 π ). • ϕ is the angle between the positive zaxis and the line segment connecting P to the origin (so 0 lessorequalslant ϕ lessorequalslant π ). 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 ( ̺, ϑ, ϕ ) in 3 space are x = ̺ sin ( ϕ )cos ( ϑ ), y = ̺ sin ( ϕ ) sin ( ϑ ) and z = ̺ cos ( ϕ ). ( 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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This note was uploaded on 10/28/2010 for the course MA 261 taught by Professor Stefanov during the Fall '08 term at Purdue.
 Fall '08
 Stefanov
 Calculus, Integrals

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