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 Integral Calculus Made Easy
Name: Phoebe L.
Status: student	
Age:  N/A
Location: N/A
Country: N/A
Date: N/A 

What is the definition of integral calculus -- easy enough so kids can understand, please.

Here is the basic idea: If you know where something is at every instant in time, you ought to be able to figure out how fast it was moving at any instant, and how rapidly it was accelerating, etc. Calculus is the tool you use to do it.

Tim Mooney

Integral calculus is essentially looking for the area made by a curve in a certain coordinate system. Assuming that one looked at a rectangle, with the long axis situated on the x axis-in a two dimensional coordinate system. Let us say that the width of the rectangle is 5 units and the height of the rectangle 10 units (along) the y-axis. What is the area of the rectangle? 5 x 10 or 50 units squared. We could also divide the rectangle into sections along the x-axis and sum up all the parts of the rectangle. OK, let us divide into 5 rectangles of width 1, then the sum of all the component rectangles would be 1x10 + 1x10 +1x10 +1x10 +1x10, 50 once again. One can continue the process and make the x axis-lengths that you are summing over (integrating) smaller and smaller. This is an easy example since the function is a constant y=10, but what if the y axis function is a parabola? Then one would have to take very small increments and sum them up to get the area under the curve. A parabola is y=x squared, the integral would be x cubed/3. This would yield an area of 5x5x5/3. Good luck.

Dr. Harold Myron

Well it depends upon the age and the skills of the "kids" how to answer the question appropriately. Let us start with an easier question, "What is differential calculus?" Differential calculus is a set of mathematical theorems that allows one to find the slope of a curve at any point on the curve, provided certain conditions are met. One can "sell" the idea of "good behavior" as "smoothness" of the curves. I think most kids even young ones will accept those restrictions. The relevance of being able to do so can be explained in terms of knowing "how fast" something described by a function is changing. When it has a maximum or minimum. How if you know the "slope" of a curve you can calculate a line that is perpendicular to the curve at that point. You can even explain qualitatively second derivatives as how fast the slope is changing and that has to do with how "tight" the curve is. Having gotten the kids "on board" with differential calculus, you can explain that integral calculus is the reverse process. That is, often we know "how fast" something is changing, and want to find out the formula (function) for the curve that changes according to "how fast" it is changing. Explain that this is a more difficult thing to do, and a lot more things have to be developed to do this. Without going into the "plumbing" you can say that finding the areas, and volumes of various shapes is done using integral calculus, as well as the length of curves (pieces of string that are curved), the surface areas of various shapes. I am assuming that you are talking to fairly young kids, so I think the way to "explain" calculus is to describe all the "neat things" you can calculate using calculus.

Vince Calder

Phoebe L.,

Integral calculus is the portion of math that calculates the total effect of many tiny pieces added together. It tells the total change based on many small changes.

Dr. Ken Mellendorf
Physics Professor
Illinois Central College

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