Triple Integrals over General Regions

Triple integrals over general regions describe accumulation across a bounded solid in three-dimensional space. The region E may have curved or irregular boundaries, so a systematic way is needed to define the bounds.

To define the integral over E, the solid is placed inside a larger rectangular box. A new function matches the original function inside E and is zero outside the solid. This defines the triple integral over E using the surrounding box. But for practical evaluation, the solid can be described using vertical slices.

A practical example is estimating the amount of ore in an underground deposit whose top and bottom surfaces are given by two functions.

The projection of this deposit onto the horizontal plane forms a base region, D. Each point in D gives a vertical segment through the solid, from the lower surface to the upper surface.

The triple integral is then evaluated as an iterated integral using vertical slices. In the inner integral, values are added along the vertical segments. These totals are then added over the base region D.

So, the triple integral gives the total amount of ore within the solid.