15.8 - Section 16.8 Triple Integrals in Spherical...

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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 x -axis, so 0 lessorequalslant ϑ < 2 π ). ϕ is the angle between the positive z -axis 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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