Applications and Approximations of Multiple Integrals
It is customary in one-variable calculus to include geometric applications of Riemann integration so as to give definitions and methods for the evaluation of area of a planar region, arc length of a curve, volume of a solid of revolution, and area of a surface of revolution. (See, for example, Chapter 8 of ACICARA.) While the definition of arc length thus obtained is quite general, the definitions of area, volume, and surface area are applicable only to a restricted class of planar regions, solids, and surfaces. In fact, general definitions are obtained using the notions of double integrals and triple integrals developed in Chapter 5. In Section 6.1 below, we discuss the general notions of area and volume, and show that these are consistent with the definitions given in one-variable calculus for certain regions in R2 and solids in R3. Areas of surfaces in R3 are discussed in Section 6.2, and it is shown that areas of surfaces of revolution are a special case. Subsequently, a general treatment of centroids of planar regions, solids, and surfaces is given in Section 6.3, and this includes a theorem of Pappus relating the volume of a solid of revolution with the area of the corresponding planar region and its centroid. In the last section of this chapter, we consider cubature rules, which are higher-dimensional analogues of quadrature rules given in Section 8.6 of ACICARA. These are useful in finding approximations of double and triple integrals.
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