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Name: Greg M.
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Question:
Last semester in Calculus III, my professor, who does research in topology, mentioned hyper-spaces and 0-dimensional objects along with infinite dimensional objects. I understand that when one evaluates a triple integral of a function of three variables, he or she can calculate the mass, volume, moments of inertia, centroids, etc., but do these functions have any geometrical interpretation besides applications in physics?



Replies:
Greg,

Multi-dimensional integrals can have a wide variety of geometrical interpretations. It relates so often to physics because physics is the science that relates to EVERYTHING real. You can use multi-dimensional integrals whenever the change of one quantity depends on changes of more than one INDEPENDENT parameter. Such relations can exist in finance, weather, chemistry, biology, medicine, and many other fields.

One example that comes to mind is population density. It is possible to have a mathematical approximation for number of people per square mile within a city, as a function of both North/South and East/West coordinates. Integrating this density with respect to both of these coordinates, over a specified area, yields a very good estimate of the number of people living within that area.

Dr. Ken Mellendorf
Illinois Central College


The "integral symbol" remember is a shorthand for a SUMMATION of some sort in which the variables can be considered to be at least piecewise continuous -- at least ignoring some special cases. So any application where there is some sort of summation that is piece-wise continuous, an integral may pop up. They occur in diverse fields from astronomy to zoology A to Z -- and everything in between, for example: chemistry, economics, biology, etc. The interpretation of the "meaning" of an integral depends upon the discipline. That interpretation may or may not have any geometric meaning. The important thing to remember is that an integral is fundamentally some sort of summation.

Vince Calder



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