In this section you are going to study one of the most remarkable relationships in all of mathematics. Some historians regard this discovery by Newton and Leibniz as the most important event in the history of science. Sounds a little deep, doesn't it, but if you look at what occurred in the modeling of the natural laws of science after the discovery, its importance becomes very clear.
This relationship is the fact that the area problem and the slope problem are intimately linked to each other. Although the concept can be somewhat difficult to understand at first, the application isn't. In this section you will look at a simple, but practical example that should bring the whole concept in to focus.
In the last section you found that the area under a positive continious function gives you valuable information, which on the surface does not seem to pertain to the variables chosen in the plot. For example, in the volocity formula you studied in Chapert 2, you plotted time against velocity. You input time and the graph gave you back velocity.
In Part 1 of this section you found out, however, there was much more information in this simple graph than first realized. You found that you could find the distance traveled by finding the area under the graph for the time given. To do this you learned to inscribe ever thinner rectangles under the curve totaling up all of their values to come to a close approximation of the area.
For some functions you do not even need calculus to find the area under the plot. For example, any constant function is very easy to determine; it is merely a rectangle that you find the area of. The function itself is the height and the chosen domain is the the width. Below are a couple of other functions that have easy formulas where you can determine the area under the plot.
What follows is a concept that is very simple in its implementation, but profound in its implication.
From very early in your mathematical training, you have been able to find the average of a list of numbers. You total n amount of numbers and divide that total by n to find the average. Little did you know that you were finding the average of a function; the function described by that set of numbers.
Take for example the following: You measure the temperature outside your home each hour for twenty-four hours. The following table lists those numbers (you start at 1 am). The first set is the list in the form of a spreadsheet and the second is the LiveMath table you get after you Paste the list into a Notebook. The Table is given the User Defined function name "T".
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You can find the average temperature by one of two methods. First, you can total the list on the left and divide by 24.
Using LiveMath, you can also use the Sumation Op on the table data using the following steps. First, sum all of the temperatures in the table.
The summation process merely places the counter k into the function T, extracts the number associated with that counter, and cumulatively adds all of the temperatures up. The same total (283) is achieved.
To find the average, you divide by the number of temperatures; in this case you took 24 readings.
The graph of this experiment is shown below with the temperature graph (Temp) in black and the average of the function (11.792) in red.
What does this Average number mean? Look at the next graph which colors the area under the Average line.
The area under this red line (green area) is equal to the heigth (11.792) times the width (24).
This is interesting! The area under the Average Function is the same as the sum of the total temperatures. Since the temperatures were taken one per hour, in other words, the delta-t was 1, the total (283) is also the area under the the Temp curve.
Why? Take a look at the following. The sumation you used to total all the temperatures is similar to the Riemann formula you derived in the last section. All that is missing is the delta-t, but delta - t is equal to 1 in this example. Therefore they are, for the most part, the same.
when
delta - t is equal to 1.As a practical example, think of the black line in the graph as a cross section of a pile of snow bound by the domain of the function (0->24). If that snow would melt, the water (green) would come to the level of the red line (the average).
Putting this all together into a general mathematical form, you have the following equation defining the average value of a function.
As a definite integral:
When you look at the average function, you see that to find the area of this rectangle, you merely take f(b) times the interval (b - a). In the example above, f(b) = 11.792 and b - a == 24 -0 = 24. And 24 * 11.792 ~ 283.
Back in Chaper 2, you found the slope of a function at various x-values. You then plotted these slope values and found that the function defined by these points was a new function defined as the derivative.
You then plotted this function in the same Graph Theory that held the original function. To find the slope of the original function, you just had to extract the y-value off this new function at the given x-value.
Later, you may have looked at the characteristics of this new function as it pertained to the original function. For example, when the derivative function crossed the x-axis, you noticed that the original function reached a maximum or minimum.
Why not use the same general concept, but this time reversing it, to study the integral.
You know that you can determine the area under the graph of a positive continious function by using Riemann Sums. By subdividing the special interval ( 0 -> b ) into equal partitions and calculating the cumulative area under the curve at each x-value, you can generate a table of numbers.
Why not plot these numbers on the same graph of the function you are trying to integrate. Below is a movie of just this.
The original function is the simple linear function f(x) = 2 x. The subinterval is chosen to be 0.5. It is obvious with this simple example that the plot of these points representing the area under successive x-values is the squaring function; y = x^2. This function, as you discovered in the section on anti-differentiation is the anti-derivative of y = 2x.
As you tab through the movie, notice how each point gives the TOTAL area of the red triangle at each new value for x. You can verify that these numbers represent the area by merely finding the area of the red triangle (1/2*b*h) at each point.
What this says is the following: The area under a positive continious curve from a = 0 to b = x is equal to the value of the anti-derivative of that function at b (the upper bound).
In the example above, all you need do is substitute x into the function x^2 and the answer will give you the area under the red curve from 0 to that value of x. The movie above does this for you. Look at the "Red Area" values. Each one comes from putting the value of x into the anti-derivative function x^2.
Below is the math:
Using LiveMath's Definite Integral Op, look at how the answer is merely the function's value at the upper bound b.
When you define the upper bound (for example x = b = 4), then the solution is:
You may have the following question. What if you want to find the area under a function when a is at some other point than zero (0). This problem is solved very easily when you think of this little metaphor. Let's say you have a string that is 12 inches long, but need one that is 9. What do you do? You cut off the first 3 inches. This is basically what you do to find the area under curves that do not start at zero.
You have a string 12 inches long (the f(b) length). To obtain a string 9 inches long, you subtract 3 inches (the f(a) length).
Rather than take an easy linear equation, this time use the following:
You want to find the area under the given function from a = 2 to b = 4.
First you would find the anti-derivative of the function using the Power Rule. Below it is plotted it in the same Graph Theory (Blue line).
From above, you know that the first thing you do is find the area from 0 (zero) to b (Plot #1 below), You would then find the area from 0 to a (plot #2). Note: the graph bounds have been changed in the following Graph Theories to better isolate the domain.
The third graph shows the resulting area (red) that is achieved when you subtract Graph #2 from Graph #1.
The mathematics for the above is quite simple too. You find the area of the whole area under the plot from 0 -> b (Graph #1 Red Area=). Then you subtract the area under the plot from 0 -> a (Graph #2 GreenArea=). The Notebook shown below uses the Mid-Point method of finding the area.
LiveMath allows you a simple way of using the definite integral to find a more accurate answer. As long as the function you are studying has an anti-derivative, you can use the definite integral in the following manner. Remember, normally a definite integral in LiveMath will sum up the many rectangles under the curve according to its internal algorithym. By turning off Auto-Simplify, however, Livemath walks you through the implementation of the Fundamental Theorem.
After you substitute the function into the integral, you turn off Auto-Simplify and then apply the Simplify command. This generates the Evaluation Op and by continuing the simplification process, LiveMath solves the integration and applies the upper and lower bound to the resulting equation. It then subtracts the results to give the final answer.
Click here to see a Plug-in example which gives you the oportunity to change the domain of this example to see a graphical display of the concept.
Mathematically the Fundamental Theorem is presented as:
What is this F ' (x) = f ( x ); You ask? Click here to bring up a page that derives the Fundamental Theorem for you.
Below is a graphic displaying the Fundamental Theorem and its analytical implications. Note that this graphic is from the viewpoint of the integral where f(x) is the function you start out with. F(x) is the anti-derivative function.
Below, b = 4 so f(b) = f(4) = 635 which gives the slope of F(b). The slope is represented by the red tangent line.
Also, F(4) = 908 which is the area under the curve of f(x) (Green area labeled A(x)) from zero 0 to 4
Click here to see another practical example on the Fundamental Theorem.
