Integration of Functions of Three Variables
نویسنده
چکیده
In the study of functions of two variables f(x, y) defined over a region in the plane, the double integral has two interpretations: the signed volume between the graph of z = f(x, y) and the xy-plane or the total mass of a lamina with density determined by the function f(x, y). In the case of a function of three variables f(x, y, z), the graph w = f(x, y, z) is a subset of four-dimensional space, and the anologue of the interpretation of the double integral as signed volume is the interpretation of the triple integral as signed hypervolume under the graph. This is not as easy to visualize as its analogue is in three-space. However, there are other uses of the triple integral. For example, we can still consider a region in three-dimensional space with a density f(x, y, z) and look at the triple integral of the density function over the region as the total mass of the region.
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