Triple Integrals in Cylindrical Coordinates

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



and conversely,



    A triple integral can sometimes be evaluated by transforming to cylindrical coordinates if the region of integration is -simple and the projection of onto the -plane is a region that can be described more naturally in terms of polar coordinates over the region of integration   
    

Proposition (Triple Integral in Cylindrical Coordinates) Let be a region with upper surface and lower surface and let be the projection of the solid onto the -plane expressed in polar coordinates. Then, if is continuous on the triple integral of over is given by

Example (Triple Integral in Cylindrical Coordinates) Find the volume of the solid bounded by the paraboloid and the -plane.

    Solution. In the -plane we have the region Thus in cylindrical coordinates we have,
    

Example (Triple Integral in Cylindrical Coordinates) Use cylindrical coordinates to compute the integral



where is the solid bounded above by the plane and below by the surface

    Solution. We consider the region of integration as being -simple by projecting onto the -plane; and in the -plane we have the region bounded by

    

and so in cylindrical coordinates we have,
    

Example (Triple Integral in Cylindrical Coordinates) Use cylindrical coordinates to compute the integral



where is the cylindrical solid with

    Solution. We consider the region of integration as being -simple by projecting onto the -plane; and in the -plane we have the region   Noticing that



In cylindrical coordinates we have,
    

Example (Triple Integral in Cylindrical Coordinates) Use cylindrical coordinates to compute the integral



    Solution. The region of integration is





In cylindrical coordinates we have,
    

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Triple Integrals In Cylindrical Coordinates
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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