Need Help With Integration Concept

[rayroshi]

A little background: I am trying to learn calculus on my own, and am doing fairly well, actually, considering the very slow pace at which I am progressing and the fact that I am 70-years old, lol! I am working my way through Mark Ryan's book, "Calculus for Dummies," which is a very good book. I am about 3/4 of the way through, and feel pretty confident that I understand everything, so far; however, I am stuck on a concept found on page 240, concerning integration. This concept is one which is marked as being a critical one for understanding calculus, so I am concerned, because it seems to be lacking something. I realize that any "lack" is probably within my own understanding, and not in what he has said, so I am now reaching out for some help/clarification.

In the previous two pages, leading up to the statement in question, he shows how integration is simply the process of sweeping out an area beneath a function on a normal x-y Cartesian-coordinate graph. To illustrate, he proposes that the reader visualizes the function as being a straight line, as in the case of a constant, at a height of 10 on the y-axis. Then, for each increase in the x axis, the function will sweep out an additonal area of 10 units: move from 0 to 2 on the x-axis and you've swept out 20 units of area below the "curve," move from 2 to 5 on the x-axis, and you've swept out an additonal 30 units of area, etc. So far, so good; that's pretty obvious to anyone, I'm sure, and doesn't even involve calculus. No brainer.

But then, on page 240 is where he makes the statement which has me stymied: "The--rate--of area being swept out under a curve by an area function of a given x-value is equal to the--height--of the curve at that x-value." As stated, this concept is marked as being one critical to the understanding of calculus, and leads, in only one more paragraph, directly to the Fundamental Theorem of Calculus.

So here is my question: how can the--rate--of the area being swept out equal the height? Rate, I would think that rate is how fast, per unit time, the x-number is being drug across the x-axis to the right, and would have nothing to do with the height. Rates always involve time, otherwise they wouldn't be called "rates."

To be sure, the height obviously affects the amount of area being swept out, but I don't see what it has to do with how fast the x-number is being drug to the right, across the x-axis. How can the rate simply equal the height? Help!

 

[galactus]

There is a direct link between derivatives and integration(antiderivatives). Think of them as being opposites of one another.

If F(x) is the antiderivative of f(x) and C is a constant, then F(x)+C.

Since differentiation is the rate of change, then by taking the derivative of the antiderivative, we get back to f(x) which is the height they are speaking of.

Now, if we take the derivative of F(x), \(\displaystyle \frac{d}{dx}[F(x)+C]=\frac{d}{dx}[F(x)]+\frac{d}{dx}[C]=f(x)+0=f(x)\)

Since \(\displaystyle \frac{d}{dx}[F(x)]=f(x)\) then functions of the form F(x)+C are the antiderivatives of f(x).

We denote this by writing \(\displaystyle \int f(x)dx=F(x)+C\)

I hope I didn't manage to confuse you more.

 

[JeffM]

Ray

Don't worry: the INTUITION behind integral calculus is elusive. The really deep thought behind calculus is studied in a course called, at least in the US, Analysis, a topic by the way that I did not study so this may be the blind leading the blind.

Let's consider F(x) as the function for which you are interested in finding the area between the graph of F(x) at x = 0 and any other value of x and the x-axis. The resulting area is represented by a DIFFERENT function; let's call it A(x).
Now the rate of change of A(x) = dA(x)/dx = A'(x) = the limit as h approaches 0 of [A(x + h) - A(x)]/h.
Now consider the area between x and (x + h) under the curve F(x). Obviously it is A(x + h) - A(x). Is this beginning to look familiar? Now imagine that h is small. So, the area is approximately equal to a rectangle with width h and height F(x), which has an area of h * F(x). So, approximately,
h * F(x) = A(x + h) - A(x), which means, approximately, F(x) = [A(x + h) - A(x)]/h.
Now it can be shown in Analysis that the limit as h approaches 0 of [A(x + h) - A(x)]/h = A'(x) = F(x) if F(x) has certain general properties. So F(x), assuming it has the necessary properties, is the rate of change of A(x).

If this seems complete nonsense, please tell me where you think it degenerates into nonsense, and I'll try to clarify as best I can. (I am not a mathematician; my academic training was in European languages and history.)