The last bit of multivariate calculus we need to address is integration
over *multiple* dimensions. We will have many occasions in this
text to integrate over *lines*, over *surfaces*, and over *volumes* of space in order to obtain quantities. The integrals
themselves are not difficult - in this course they can *always* be
done as a series of one, two or three ordinary, independent integrals
over each coordinate one at a time with the others held "fixed". This
is not always possible and multiple integration can get much more
difficult, but we *deliberately* choose problems that illustrate the
general idea of integrating over a volume while still remaining
accessible to a student with fairly modest calculus skills, no more than
is required and reviewed in the sections above.

[Note: This section is not yet finished, but there are examples of all of these in context in the relevant sections below. Check back for later revisions of the book PDF (possibly after contacting the author) if you would like this section to be filled in urgently.]

Robert G. Brown 2011-04-19