Volume Under a Twisted Plane

If you want to know what a twisted plane is, check out my previous post on it.  Basically, a twisted plane is what you get if you take a plane and twist it and the application I have in view is surface modelling.  In that previous post, I developed the equation of this surface as defined by the (x, y, z) coordinates at the 4 corners of a rectangular area (rectangular in plan view).  For review, that equation is

In construction estimation, it is often necessary to measure volumes of material to be excavated or filled in.  To do so, we need to be able to turn elevation and position data into volumes.  A common approach is to find average depths and multiply by the relevant area.  If the depths are different in different areas, the average will need some kind of weighting or the areas treated separately and summed afterwards for a total.  If using a TIN to model a surface, volume calculations are straightforward and fit this idea of using an average depth and multiplying by the area.  (I have the derivation here.)

So, an interesting question that remains, is whether the twisted plane is as well behaved.  The answer is yes, the derivation is straightforward (but tedious to write and is omitted), and the result is

(The result is obtained by integrating z(x, y) for x over [0, length] and then y over [0, width].)