A Course in Analysis by Niels Jacob & Kristian P Evans

Author:Niels Jacob & Kristian P Evans
Language: eng
Format: epub
Publisher: World Scientific Publishing
Published: 2016-05-22T04:00:00+00:00

Now with yjk := Fk(xj), 0 ≤ j ≤ N, 0 ≤ k ≤ M, we get a partition of V into sets

for 0 ≤ j ≤ N, 0 ≤ k ≤ M, to which we must add the sets Γ(f2) as well as {b} × [f1(b), f2(b)]. For notational purpose we set

so that Vjk ∩ Vlm = for (j, k) ≠ (l, m) and .

Figure 16.8

For the following we assume that VN+1M, VNM+1 and VN+1M+1 to have volume 0 and we will not need to take care on these sets when discussing the volume of V or integrals for functions defined on V.

We can look at Figure 16.8 from two different points of view. It may serve us to illustrate once more Cavalieri’s principle (for a set in the plane) but it may also lead to ideas how to construct partitions of sets in the plane. Both views are however related.

Introducing the sets

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