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The synonymous prototype forms of volume integral functions in 2D are:

 

INTEGRAL ( integrand, region )

VOL_INTEGRAL ( integrand, region )

 

Here region can be specified by number or name, or it can be omitted, in which case the entire domain is implied.

 

In two-dimensional Cartesian problems, the volume element is formed by extending the two-dimensional cell a single unit in the Z-direction, so that the volume integral is the same as the area integral in the coordinate plane.

 

In two-dimensional cylindrical problems, the volume element is formed as 2*pi*r*dr*dz, so that the volume integral is NOT the same as the area integral in the coordinate plane. For the special case of 2D cylindrical geometry, the additional operator

 

AREA_INTEGRAL ( integrand, region )

 

computes the area integral of the integrand over the indicated region (or the entire domain) without the 2*pi*r weighting.