By Choquet-Bruhat Y., DeWitt-Morette C., Dillard-Bleick M.

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U, v). Here we have four cases: a) Rectangular plane integral. b) Polar plane integral. c) General transform. d) Surface integral. These cases are treated one by one in the following. a) Rectangular plane integral. Formula: A f (x, y) dx dy. domain of integration = parametric domain. Measure theory: Weight = 1. Comment: Basic form. Further reductions are obtained by means of Chapter 10. com 58 Calculus 2a The setting up of a line, a plane, a surface or a space integral b) Polar plane integral.

A B 5) If B (or a sub-domain of B) is of type 2, then write instead B = {(x, y) | c ≤ y ≤ d, X1 (y) ≤ x ≤ X2 (y)}, and continue with the reduction formula d X2 (y) f (x, y) dS = f (x, y) dx dy. X1 (y) c B According to the ﬁgure of the second version one starts by keeping y ﬁxed and integrate horizontically after x. This can be interpreted as collecting the “mass” over den horizontal line in the corresponding point y on the y-axis. Afterwards the total“mass” is obtained by integrating the partial results after y.

Then go to 4). 4) Next turn to A− . If A− = ∅, then go to 5). If A− = ∅, choose one of the trimming possibilities 2 − a) Rectangular: A− n = [−n, n] ∩ A , − b) Polar: A− R = K(0; R) ∩ A , and calculate a) Rectangular: In− = A− n f (x, y) dS, − b) Polar: IR = A− R f (x, y) dS. com 54 Calculus 2a Improper integrals; unbounded domain − If In− → −∞ for n → +∞, resp. IR → −∞ for R → +∞, then the improper integral is divergent. Otherwise, a) Rectangular: b) Polar: A− f (x, y) dS = limn→+∞ A− n f (x, y) dS.