1 Words ‘up’ and ‘down’ are used here in the sense of unknown and familiar, as in ‘downtown’, not to mean that the domain of a problem is necessarily going to be wider than [−1, 1].
2 Notice how it takes care of a singularity, if it happens to be integrable.
3 It is the same problem, in essence, as the one appearing in the exercises of the previous chapter.
4 The generic term ‘volume’ is used here to keep the discussion independent of the dimension of the problem. In a double integral it will be area, an ordinary volume in a triple integral and so on.