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Notice that b) is reduced to a). c) General. Formula: A f (x, y) dx dy = B f (x(u, v), y(u, v)) ∂(x, y) du dv. ∂(u, v) Geometry: domain of integration = parametric domain. Measure theory: Weight = Comment: This is a brief description of a part of Chapter 17. ∂(x, y) . ∂(u, v) d) Surface integral. Formula: A f (x, y, z) dS = B f (r(u, v)) N(u, v) du dv, where the surface F is given by (x,y,z) = r(u,v). Geometry: Surface = parametric domain. Measure theory: Weight = N(u, v) . Comment: This is a brief description of Chapter 16, where several special cases are described.

Besides being a manager in the Manufacturing IT department, Kim performs triathlon at a professional level. ‘NNE Pharmaplan offers me freedom with responsibility as well as the opportunity to plan my own time. com NNE Pharmaplan is the world’s leading engineering and consultancy company focused exclusively on the pharma and biotech industries. NNE Pharmaplan is a company in the Novo Group. com 55 Calculus 2b 20 The setting up of a line, a plane, a surface or a space integral The setting up of a line, a plane, a surface or a space integral When we set up an integral, we have two possible approaches: 1) A geometrical analysis.

Com 43 Calculus 2b 15 Line integrals Line integrals This is a 1-dimensional case where the domain of integration is not an interval, but a curve in the plane or the space. The reduction formula becomes b f (x) ds = K f (r(t)) r (t) dt. 5 1 1 Figure 11: The curve of parameter representation r(t) = (cos t, sin t, t). Procedure: 1) Write down a rectangular parameter representation for the curve K: In the plane: (x, y) = r(t), In the space: (x, y, z) = r(t), t ∈ [a, b]. t ∈ [a, b]. 2) Calculate the curve element (the weight is r (t) ) ds = r (t) dt.

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