Triple Integrals in Spherical Coordinates

Another useful coordinate system in three dimensions is the spherical coordinate system.The spherical coordinates  of a point P inspace are shown in Figure below, where  is  the distance from the origin to   is the same angle as in cylindrical coordinates, and  is the angle between the positive z-axis and the line segment 0P. Note that

The spherical coordinate system is especially useful in problems where there is symmetry about a point, and the origin is placed at this point. The relationship between rectangular and spherical coordinates can be seen from Figure below. From triangles  and  we have

But   and , so to convert from  spherical to rectangular coordinates, we use the equations

Also, the distance formula shows that  We use this equation in converting from rectangular spherical coordinates.

EXAMPLE 1: The point   is given in spherical coordinates. Plot the point and find its rectangular coordinates.

Thus the point  is  in rectangular coordinates 

EXAMPLE 2: The point  is given in rectangular coordinates. Find spherical coordinates for this point

(Note that  because .) Therefore spherical coordinates of the given point are .