Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne Application of Integrals 2001227
Name: Greg M.
Status: student	
Age:  N/A
Location: N/A
Country: N/A
Date: N/A 

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?


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

Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (, or at Argonne's Educational Programs

Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory