

Application of Integrals 2001227
Name: Greg M.
Status: student
Age: N/A
Location: N/A
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Date: N/A
Question:
Last semester in Calculus III, my professor, who does
research in topology, mentioned hyperspaces and 0dimensional 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,
Multidimensional 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 multidimensional
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 piecewise 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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Update: June 2012

