Multiple Integrals
Multiple integrals are a set of integrals that are taken over more than one variable. In general the more integrals that are taken the higher the dimension of space that is involved in the area's shape. For instance if two integrals comprise a multiple integral then the outcome corresponds to an area, three integrals correspond to a volume. A multiple integral generally appears as: f(x, y, z) dx dy dz. To compute a multiple integral one begins from the most inner integral and evaluates that one first, then proceeds to the next inner integral and so on. In the example above the most inner integral would treat x as the variable whilst y and z are taken as constants. The order in which one carries out computations can be varied but care must be exercised to correctly transform the limits to correspond to the proper integral. Just as with a single integral, a multiple integral is defined as the limit of a Riemann sum.
Multiple integrals arose when mathematicians confronted functions that depend on more than one variable. It was determined that in order to integrate with respect to two different variables that one simply integrated with respect to one of the variables first, then integrated the result with respect to the other variable. While integrating with respect to one variable the other variables are treated as constants. Guido Fubini, an Italian mathematician, concentrated his studies on expression of surface integrals in terms of two simple integrations. He is the author of Fubini's theorem which establishes a connection between multiple integrals and repeated ones, integrals that are several times over a single variable.
Multiple integrals can be integrated via a summation method or numerical integration methods. A double integral can be thought of as kind of a double sum. Integrating a function with respect to multiple variables involves integrating over a range of one variable and then summing the result of this integration over a range of the other variable. In this case, with two variables, the result in an area in a plane rather than just a section of one axis, which corresponds to a single integral. The trapezoidal approximation as well as other quadrature methods, such as Simpson's rule, can be applied to multiple integrals just as to single integrals. This is accomplished in the same way, i.e. first the numerical method is applied to the innermost integral and after evaluation it is applied to successive integrals working outward.
On occasions it is sometimes useful to use cylindrical and spherical coordinates when calculating multiple integrals. These substitutions often make evaluation of the results much simpler. In triple integrals in order to convert to cylindrical coordinates substitutions for x, y and z must be made: x = rcosΘ, y = rsinΘ, z = z.. After such substitutions have been made the volume element is given by: dV = rdzdrdΘ. For converting a triple integral to spherical polar coordinates the following substitutions need to be made: x = s sinΘcosΦ, y = s sinΘsinΦ, z = s cosΘ and the volume element is given by: dV = s2 sinΘ ds dΘdΦ. These substitutions often make the evaluation of triple integrals generated by rotation of a curve around an axis or line much simpler.
Multiple integrals are found in several areas of science and engineering. They are used to determine the area of a two-dimensional region, volume, mass of two-dimensional plates, force on a two-dimensional plate, the average of a function, the center of mass, the moment of inertia, and surface area. Chemistry and physics readily employ multiple integrals in quantum mechanics as well as to determine optical cross sections.
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