The Double Integral

The definite integral accumulates along an interval; the double integral accumulates over a plane — in essence it is nothing more than “accumulating once more in another direction”. The technical detail that this simple idea brings is far more than one might imagine.

This chapter first establishes the concept and the properties, paying particular attention to the conditions of integrability and the geometric meaning: the value of a multiple integral is the volume of the cylinder with curved top, and this intuition should run through the whole chapter. As for the computation, you have to master both the rectangular and the polar methods, and understand when to switch to polar coordinates — when the region is a circle or has rotational symmetry, the simplification is usually dramatic. Changing the order of integration is another key technique, and this chapter explains when it is necessary.

The section on symmetry and parity gives a large number of “no computation needed” results: if the integrand is an odd function of some coordinate and the region is symmetric, the integral is directly zero. Judgements of this kind save a great deal of work in actual computation.