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Answer 1

$<b>Double integration is the procedure that produces an [ordinarily] explicit mathematical expression for the area under a surface. Double integration [obviously] arises out of the application of two integral operators. What differentiates it from triple integrals, is that it integrates over two variables -- one integral corresponding to each variable and its respective limits of integration -- whereas triple integrals do so over three variables. In educational contexts, the double integral is introduced as the higher-dimensional analogue of the ordinary integral. Where, you'll recall, the latter [assuming it's a definite integral] can be used to evaluate the area under a curve [where a curve is one-dimensional in the sense that it it has only the physical dimension of the metric (i.e., a length), the physical interpretation of the former is the area under a surface [where a surface exists as the construction of some general planar-like region along two physical dimensions]. This is geometrically visualized using a three-dimensional Cartesian plane for which divisions are now spaces called octants while in xy-Cartesian planes the divisions are areas called "quadrants" [since you can only divide a plane into 4 sections using rectangular coordinates -- the standardized coordinate frame of Descartes' and which requires that the basis vectors (colloquially, the x- and y-axes) are orthogonal]. It is sometimes said that double integration returns area and triple integrals volume, but that is not quite accurate. For example, even single integrals can be used to find volume. If the object of interest whose volume we intend to find has sufficient conditions of symmetry, the disc method (also known as the method of rings) and/or the shell method (or the method of [representative] cylinders) can be used. We refer to these objects as solids of revolution. Calculating their volumes can be executed by taking a cross section, or slice, from the object and integrating around an/the axis of revolution. For ones with excisions through the middle, a specialized type of the disc method can be used called the method of washers [since washers, like donuts, are annuli]. Specific applications of double and triple integrals vary. They are used widely in crystallography, for example as well as an array of specific topics in most of the disciplines of the natural and physical sciences.$

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