Triple Integrals in Spherical Coordinates

Triple Integrals in Cylindrical and Spherical Coordinates


Cylindrical Coordinates

Cylindrical coordinates extend polar coordinates to 3D space. In the cylindrical coordinate system, a point P in 3D space is represented by the ordered triple . Here, r represents the distance from the origin to the projection of the point P onto the x-y plane, is the angle in radians from the x axis to the projection of the point on the x-y plane, and z is the distance from the x-y plane to the point P.


Cylindrical coordinates are good for graphing surfaces of revolution where the z axis is the axis of symmetry. One method for graphing a cylindrical equation is to convert the equation and graph the resulting 3D surface.

Spherical Coordinates

Spherical coordinates represents points from a spherical “global” perspective. They are good for graphing surfaces in space that have a point or center of symmetry.

Points in spherical coordinates are represented by the ordered triple



where is the distance from the point to the origin O, , where is the angle in radians from the x axis to the projection of the point on the x-y plane (same as cylindrical coordinates), and is the angle between the positive z axis and the line segment joining the origin and the point P . Note .