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.]