Integrals

Integrals are the running total of the function. To compute a running total, you must specify a start point and an end point - integrals start at a, and end at b. The other, common way to look at an integral is the area under the curve of a function.

Details

Integrals can be seen as the sum of all the function's values. However, a continuous function has an infinite number of values between points a and b. Shouldn't the integral be infinity?

Ordinarily, yes (this leads to things like Zeno's paradoxes). However, if we use infinitesimals we can see how an infinite number of infinitesimals can be a real number.

The secret: Let dx be an infinitesimally small number. We will sample the function at intervals of dx, and add up those samples. Another way to look at it is the sum of rectangles (a Reimann sum) -- we find the area under the curve by adding up little rectangles. The important point is that dx is small.

Why is the integral of x equal to (x^2)/2?

[needs to be updated -- I need a better explanation]

Here is a math derivation. True, perhaps, but not enlightening. Here's an intuitive answer:

If we have a graph of y = x, we get a straight line. One way to find the running total is to actually add up all the numbers. This is represented by [*integral of x *]. This implies that we haven't actually evaluated the integral yet. We are just saying that the integral of x is, well, the integral of x.

The actual answer can be figured out. Suppose we have a line from 0 to x. The range of this function is x. Thus, if we can find the average value, multiply it by x, we get the integral. How come? Well, the average is simply the total divided by the number of samples (x).

Average = total/x

total = average * x

But the total is our integral! Thus, once we find the average we are set. The average of a sequence of linearly increasing numbers (max + min)/2. The max is x (our endpoint), the min is 0 (our start point), so the average is x/2. Thus,

total = (x/2) * x = x^2/2.

Remember that our total is also the integral, so we are all set.

What about higher powers (x^2, x^3, etc)?

We can do the same analysis for higher powers. The first few terms of y = x^2 are

0 1 4 9 16 25 (for x = 5)

The total is 55. Thus,

55 ~ average * x

(where ~ means approximately equal to)

We know the average will be some fraction of the maximum value. Why? Well, the average must be below the maximum, obviously. In addition, it makes sense that the average would rise in proportion with the maximum. Go with me here, in that the average is roughly proportional to the maximum value (because we are considering x^2 ... the same distribution, just higher maximums). Thus,

55 ~ (x^2)/n * x = (x^3)/n

(n is the proportion of the maximum that the average is (for our first example, n = 2). x^2 is the maximum value of the graph for any x we choose. x = 5 in this case).

Thus, for

n ~ 125/55 = 2.2

We know the correct answer is n = 2. We will see n approach 2 as we take more than 5 samples (5 samples is pretty crude, especially in the discrete case).

We can do similar analysis for x^3. The important thing is to see why the power increases with the integral - we are multiplying the number of samples (x) by a the average value of the function. The average value of the function is some fraction of the original function, so

integral = running total of f(x) from a to b
           = average value of f(x) * (number of samples)
           = some fraction of f(x) * x
           = (1/n) * f(x) * x

[This formula is rough, but shows you where the (1/n) constant and x come from in the integral]

n will vary based on the original function f(x). Look above for a math derivation, but you can figure it out experimentally, like we did. Surprisingly, n is always an integer for polynomial functions, like the ones we have been looking at. See the math derivation for the answer.

This makes sense: an integral should be "bigger" than the underlying function (this is not always true, as we shall see).

Why do we need dx in the integral?

Notice the dx in all integrals. This is because our approximate averages are not exactly accurate. For example, we've shown that the average value of a polynomial will always be some fraction (1/n) of the maximum value.

As we take more samples, our average becomes closer to the true average. We need an infinite amount of samples to get the true average. To take an infinite amount of samples in a finite space (from a to b), we need to have infinitesimally close spacing. Thus, dx tells us to sample at intervals of dx, rather than 1. This gives us an infinite number of samples. Therefore, the proper integral has dx. Some people see this as pure notation, but it is useful to consider this infinitesimal to be a variable we can work with. Cool, eh? :-)

What about multiplicative constants?

They don't matter. Multiplying a function by a constant multiplies the total (and average) by that constant. That's why you can pull multiplicative constants out of integrals.

Cool revelations

1. Integral is like a running total/the area under the graph.

2. The integral of a polynomial function (y = x^n) is of the form (1/n) * f(x) * x. This makes sense, because an integral (a running total) should be "bigger" than the underlying polynomial.

3. We need to take an infinite number of samples to make "our" n approach the correct one. Thus, we need infinitesimally close spacing between samples (not 1), so we use dx. It is pure notation to some, but can be interpreted physically.