The Triple Integral

The triple integral is a straightforward generalisation of the double integral: regard the integrand as a density in space, and integrate the total volume or the total mass over some region. The computation has one more level of iteration, but the core difficulty lies in how to “see” the region of integration.

Rectangular coordinates suit box-shaped regions, cylindrical coordinates suit regions symmetric about an axis, and spherical coordinates suit spherical regions. The criterion is always the shape of the region itself, and not the integrand. This chapter works through how to determine the limits and the Jacobian factor in each of the three coordinate systems — getting this step wrong is where most beginners lose marks.

The triple integral is rarely examined on its own; its value is that it provides the intuitive foundation for the change of variables in the next chapter: why you must multiply by a Jacobian factor when you change coordinate systems — the answer is right here.