Chapter 10 - Theory of Integration

The goal of this chapter is to undo all that has been done in the previous chapters on differentiation. We saw how the derivative can be used to approximate the anti-derivative, but how do we use it to give us the exact behavior of its anti-derivative?

Differentiation involves two main operations. The first is subtraction or letting a change in x, get smaller. The second operation is division to calculate the relative change of the dependent variable, f with respect to the independent variable, x. Therefore, the focus of this chapter will be on summation and multiplication.

To begin our study, let us look at the definition of the derivative of a function,

What this definition does is analyze a changing function over an infinitely small interval to calculate the rate of change of the function at that instant. By letting go to zero, the terms interval and point become synonymous such that the derivative gives is the rate of change at any point x. If we multiply both sides of the equation by dx ,where we get:

We can calculate the approximate change in the function, by replacing the infinitesimal dx with a discrete

The answer is only an approximation because the equation assumes that the rate of change or derivative is constant over the interval . For example consider the following linear function:

or is a constant. At any point on the graph of , the rate of change is reflected by the constant steepness of the graph. Now let me pose a simple question. How much does the function, , change when x goes from 0 to 4? By evaluating at 0 and 4 we get:

The new symbol [ means to evaluate the function at the two values specified on the symbol and subtract the difference.

When x changes from 0 to 4,  the dependent variable, f, changes by 8. How else could we arrive at the same answer? From the definition of the derivative we know that:

Over an interval Dx, the change in f(x), can be approximated by evaluating for the interval. Let us see how this applies to the function we were studying. We have shown that as x changes from 0 to 4, f changed from 0 to 8. Now let us find the same answer using the above equation:

The answer obtained from the derivative is exact because the rate of change of the function is constant over the discrete interval,

But what about if the derivative is not a constant but varies with x? As we shall soon prove, the total change of a function f(x) from x = a to x = b is found by taking the infinite sum of from a to b where each gives an infinitely  small change in the function, df,  over the infinitely small interval, dx.

Let us look at a function whose derivative varies with x:

As x changes the rate of change of f(x) increases. We can begin by asking ourselves what is the change in f(x),  when x changes from 0 to 6? In the graph this translates to:

For the function, , when Dx goes from 0 to 8, Df is given by the equation:

How else could we calculate the change in f? From the derivative we know that:

Since the derivative of the function changes with x, then we need to evaluate the above equation over small intervals of Dx to get an accurate answer. Remember the rate of change of f(x) is changing and is not constant over a discrete interval, Dx.

By breaking up the derivative, into 8 intervals of Dx = 1,  we can approximate the change in  of from x = 0 to x=4  by using the derivative only. A small Df of the graph is found by evaluating over each interval,

The net change in f(x) is found by summing up the individual approximations of , for each interval.

Substituting known values in:

Adding up the , we get 28. This corresponds to an error of 4 since we calculated the actual Df of f(x) to be 32 as:

To approximate this we evaluated the derivative, over small sub-intervals to calculate the net change in the function, . Relating the instantaneous rate of change to the slope of the tangent is:

By evaluating the derivative over each interval Dx we got the approximate Df of f(x) over that interval only. How could we get a more accurate result? The solution is to further divide the interval into a smaller , such that the derivative is assumed to be constant over the infinitely small interval. If we let equal 0.5, we will have sixteen intervals over which we can calculate the

You can clearly see that by further reducing the interval, the error is difference between the approximate and actual is very small. By using sixteen intervals, we reduced the error from 4 to 2. We can go on to conclude that as the interval, , goes to zero, then the net change in f(x), will correspond exactly to the infinite sum of the derivatives evaluated over each interval,

Thus the net change in a function from a to b is exactly equal to the infinite sum of its derivative evaluated over an infinitely small interval, One argument against such a conclusion is that as decreases, the error gets smaller for each individual interval however the total error remains the same. This is because as , we are adding up an infinite sum of small errors as opposed to a discrete sum of 8 or 10 for example. We have shown intuitively that this does not happen,  but how can we prove it?

Returning to the definition of rate of change between two points on a graph, separated by a distance

Notice that we are not taking any limit as . The above equation holds true for any two points on the graph of . Multiplying both sides by gives us:

If we let , then 2x is the derivative function such that:

Therefore /2 represents the error associated with calculating when we ignore it. The total sum of errors from x = a to x = b, where is:

Thus the total error for calculating the net change in is:

Remember that , therefore:

This important result confirms the fact that the sum of the derivatives evaluated over infinite times over infinitely small intervals of dx from x = a to x = b corresponds exactly to the net change in the anti-derivative, f(x). We have proven this for the case of , but how do we prove it for the general case for any f(x) and its derivative? More specifically we want to prove the following fundamental theorem of Calculus.

Here, the integral sign, , replaces the summation sign, , and represents the infinite sum of the derivative evaluated over an infinitely small interval, dx. To understand this let us take a more conceptual look at how the rate of change of a function is defined. The following formula give us the average rate of change of f(x), through any two points on the graph of f(x).

As we let , the two points, come closer together. The average rate of change converges to the instantaneous rate of change of the function at that point, x.       

For any function, , as the rate of change converges to a constant value over an infinitely small interval, . Since the instantaneous rate of change, given by constant over the interval, dx, then an infinitely small change in f(x), df, is found by multiplying the definition of the derivative by dx, where

Thus the net change in is the infinite sum of the from x = a to x-b is:

The first term converges to , while the second term, represents the net change in the function, from x = a to x = b, where . Finally the last term is the infinite sum of the derivative evaluate over each interval, dx. This infinite sum is represented by the integral sign,

This the fundamental theorem of Calculus. For example if f'(x)= 2x and we sum up 2xdx as delta x-->0 over an interval, we get the CHANGE in the anti-derivative function F(x) over that same interval, where F(x) = x^2 .