Triple Integrals with Cylindrical Coordinates

In three dimensions there is a coordinate system, called cylindrical coordinates, that is similar to polar coordinates and gives convenient descriptions of some commonly occurring surfaces and solids.

In the cylindrical coordinate system, a point P in three-dimensional space is represented by the ordered triple  where  and  are polar coordinates of the projection of P onto the xy-plane and z is the directed distance from the xy-plane to P. See Figure

To convert from cylindrical to rectangular coordinates, we use the equations

whereas to convert from rectangular to cylindrical coordinates, we use

EXAMPLE 1:

(a) Plot the point with cylindrical coordinates  and find its rectangular coordinates.

(b) Find cylindrical coordinates of the point with rectangular coordinates .

(a)

So the point is  in rectangular coordinates.

(b)

Therefore one set of cylindrical coordinates is . Another is  . As with polar coordinates, there are infinitely many choices.