The two main ideas in Calculus are the derivative and the integral. Both of these ideas are easily described geometrically in terms of the graph of a function. Let's say we've got a function y = f(x), then the derivative y = f'(x) represents the slope of the tangent line to the function at the point x. The derivative can also be thought of as the rate of change of the function f(x) at the point x. Likewise, the integral can be thought of as the area between the graph of the function f(x) and the x-axis, but in a rather awkward way in which the area formed when f(x) is negative is subtracted from the from the area formed when f(x) is positive. There is another way of thinking of the integral which may make this adding and subtracting seen a bit more natural.

Let the function represent a flow. In this particular interpretation of the integral we should think of the function y = f(x) as representing a flow. This could be a fluid flow, such as water flowing over a dam, or a flow of money into and out of a particular account, or even a flow of people into or out of a certain sociological condition, or into and out of a certain geographical location. You can use your imagination to invent all sorts of ways in which a function can represent a rate of flow. In these examples the independent variable, usually x, is probably better thought of as time and represented by t. So we think of y = f(t) as being the rate of flow of something at the precise time t. The function f(t) could be the number of dollars per year flowing into or out of an account at a given time t; f(t) could just as well represent the number of people per year flowing across the poverty line, one way or the other, at a certain moment t. It is important to keep in mind that to represent a flow, the function y = f(t) should have a unit that represents something changing over time., that is the unit should be in the form (units of something) per (units of time).

Flow takes place in two directions. Notice that in the above examples the flow is reversible. You can have a flow into something and the flow can change directions, and then you get a flow out of something. If we are saving money in an account, the money flows into the account. But if we are saving money we are probably, at the same time, also spending some money. When the money that we are spending becomes greater, per unit time, than the money that we are saving, then the flow reverses and we have a flow out of the account. The same holds true for the other examples.

When water is building up behind a hydroelectric dam two processes are taking place: some of the water is pouring through the dam generating electricity; also water is pouring into the reservoir behind the dam from the river which was dammed up. At a given moment t, if the rate of water going through the dam is subtracted from the rate of the water coming into the reservoir, the result we get is the flow y = f(t) of water into or out of the reservoir. Next we look at how the integral related to this.

The accumulation of a flow. Whenever we are working with flow it is best to think in terms of an example. Money is always a popular topic. Let's think of the flow of money into, or out of, an account. The difference between the deposits and the withdrawals is the net flow of money into or out of the account. If the net flow at time t is positive we get a positive f(t) and money builds up, or accumulates, in the account. If the net flow of money at time t is negative we get a negative f(t) and money flows out of the account. Eventually the accumulation of money becomes negative. This means that the account will go in the red. Then our account represents a debt rather than an asset.

The same type of process takes place in any example of flow. An accumulation takes place when the flow is positive. This accumulation is decreased, and may even go negative, when the flow is negative. Although, in the example of water accumulating behind a dam we cannot actually have a negative amount of water in a reservoir. However the size of the reservoir, in cubic meters of water, increases when the flow of water is positive and decreases when the flow of water is negative.

In Calculus textbooks; distance, velocity, and time, are frequently discussed. Velocity is the variable that plays the role of flow. This is because velocity has a unit which is set up in terms of something per unit time. In this case it isn't money, water or people, but distance that is being measured per unit time. Thus velocity has units like kilometers per hour. When we experience velocity the thing that accumulates is distance. Velocity is like the other flows in that it changes with time. When we are riding in a car our velocity at a given instant can be greater or less that it was before. The accumulation of distance in a particular trip depends on the whole history of the velocity flow function just as the accumulation of water behind a dam depends on the entire history of the water flow since the dam was built.

So far we have discussed the flow of people, money, water, and distance. We understand that the flow of distance is called velocity. In working with these ideas see if you can think of other flow and accumulation examples.

But what does the integral have to do with all of this? The integral is nothing more than the mathematical tool which takes care of accumulating a flow. Let's discuss this next in greater detail.

The Integral as an accumulator. When we express the integral of a function f(x) the integral sign is really just an addition sign. In the case of the integral the things being added up are not thought of a separate objects as with an ordinary plus sign (or even a sigma sign), but as one continuous substance which is flowing into the integral and being accumulated in some way or other. So the important thing here is that the integral is really an addition.

It is very important to make a clear distinction between the addition of discrete objects and continuous substances. When we add up discrete objects we use an ordinary plus sign or maybe a sigma sign. This is what would happen if you were to add up the weights of five different cans of food, or the populations of seventeen cities. Addition can of course involve negative values, and sums can themselves turn out to be negative. When we are adding up a continuous substance we think of the addition as an accumulation and we use the integral sign to do the addition.

Continuous substances don't come in separate packages, but they flow into the addition in one smooth process. The function f(x) or f(t) represents the rate of flow of the substance into the sum. A substance can gush in rapidly as happens when f(t) is great, or trickle in as when f(t) is small. When we have a negative f(t) then the substance is being lost, and the accumulation is decreasing.

You can think of the integral sign as being very much like a sigma sign. The difference between the two is that the sigma sign is for adding up separate packages, and the integral sign is for accumulating a continuous flow. It is quite appropriate that the sigma sign has a square boxy look, and the integral sign is a smoothed out version of the same thing. This smoothed out addition sign represents the flowing in, or the oozing in, or the gushing in, of a continuous substance being built up in a continuous addition.

One of the nicest aspects of integration is that the integral sign automatically takes care of addition for f(t) positive, and subtraction for f(t) negative. That is the final integral is the resultant of all the flows both into and out of the accumulated sum. There is no need to find positive flows and negative flows separately, and then do a subtraction. This makes the use of the integral operation very efficient from a mechanical point of view.

The discrete as continuous. Frequently we have applications of the integral to items which are really discrete. Examples like those involving people should really be thought of as discrete, using ordinary addition. However, when the number of people is very large the addition becomes more and more like a continuous accumulation of a flow, and we are not far wrong to use integration. It is testimony to the mechanical efficiency of integration that we tend to use it whenever we can. So we frequently think of separate items as forming a continuous substance and use integration even to add up these discrete items.

A variable fluid flow, like water, has to be added up using integration. A flow of a powdered substance, like sugar, should strictly be added up using ordinary addition. This would be so tedious, however, that we choose to think of the sugar as being just like the water. So we decide to add up its variable flow also using integration. When you work with the integral, try to decide whether the example you are doing represents a truly continuous substance or a discrete substance which is being thought of a continuous in order to use the integration process.

In summary, we should not think of the integral only as representing areas, but as being an addition or accumulation of continuous substances which flow in at variable rates given by f(t). When we think of the integral this way it is easy to see why it is so important in so many different fields. After all, addition is important whenever we deal with things that can be measured or counted. And integration is just another kind of addition.

David Gates