The concept of definite integration can be extended to two or three or multi dimensional system. For a bounded function f(x,y,z) defined on a rectangular box B (x0 = x = x1, y0 = y = y1, z0 = z = z1), the triple integral of f over B,
Integral (B) [f(x,y,z)] dV or Integral (B) [f(x,y,z)] dx dy dz,
can be defined as a suitable limit of Riemann sums corresponding to partitions of B into sub boxes by planes parallel to each of the co-ordinate planes. Triple integrals over more general domains are defined by extending the functions to be zero outside the domain and integrating over a rectangular box containing the domain.
All properties of double integrals hold good for triple integrals as well. In particular, a continuous function is integrable over a closed, bounded domain. If f(x,y,z) = 1 on the domain D, then the triple integral gives the volume of D.
Volume of D = Integral (B) dV
Triple integrals spherical coordinates: The spherical co-ordinates related to Cartesian co-ordinates x, y and z are given by the equations:
The triple integrals in spherical coordinates relationships are illustrated in the picture below
For the co-ordinate surfaces in the spherical co-ordinates, see picture below:
The volume in spherical co ordinates would be like this:
Spherical co ordinates are suited to problems involving spherical symmetry and, in particular, to regions bounded by spheres centered at origin, circular cones with axes along the z axis, and vertical plains containing the z axis.
