Triple Integrals in Spherical Coordinates

In this topic we state the usual definition of spherical coordinates and work out many examples of triple integrals in spherical coordinates. In particular we state a theorem that allows us to compute a triple integral in spherical coordinates, which among other examples, allows us to compute the volume of a sphere.

    Each point in three dimensions is uniquely represented in spherical coordinates by using and The conversion formulas from rectangular coordinates to spherical coordinates are



and conversely,



Using Jacobians, we can show that the element of volume in spherical coordinates is



Using this formula, we can form partitions and take a limit of a Riemann sums as the partitions are refined to obtain the following integral.

Proposition (Triple Integral in Spherical Coordinates) If is continuous on the closed bounded region , then the triple integral of over is given by





where is the region expressed in spherical coordinates.

Example (Triple Integral in Spherical Coordinates) Using spherical coordinates verify that the volume of a sphere is where is the radius of the sphere.

    Solution. Since the equation of the sphere is for and Then the volume is given by the triple integral in spherical coordinates,

Example (Triple Integral in Spherical Coordinates) Evaluate



where is the solid hemisphere and

    Solution. Using spherical coordinates we have,

Example (Triple Integral in Spherical Coordinates) Evaluate



where is the solid hemisphere

    Solution. Converting to spherical coordinates we have,

Example (Triple Integral in Spherical Coordinates) Evaluate



    Solution. Converting to spherical coordinates we have,

Example (Triple Integral in Spherical Coordinates) Find the volume of the solid region that remains in the spherical solid after the solid cone has been removed.

    Solution. We have

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