Integral Calculus Made Easy
Name: Phoebe L.
What is the definition of integral calculus -- easy enough so kids can understand,
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.
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
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.
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
Illinois Central College
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Update: June 2012