More on the Fundamental Theorem

Back in Chapter 2, you looked an example of a car trip to start the study of derivatives. In that example, you found that the derivative of the distance function is the velocity function. Reversing this process by finding the anti-derivative of the velocity function, you will be given the distance function. You are reversing the process.

Let's say you are traveling at a constant speed of 50 mph for a period of 4 hours. The graph of this function, f (t), is merely a straight line, as is shown in the graph below. The graph plots the velocity function.

Since the function is constant, no matter what time, t, you choose, you will always get the same speed; 50 MPH. So at hour #4 you are going 50 MPH as well as at hour #2 or #1 and so on. If you found the area under the curve (as you did in the last section), what would you have? Take a moment to think about it.

Since it is a constant curve, wouldn't you find the area by just taking the value of the function times the hours traveled? This is exactly what you do! So, if you used 4 hours as the total time, the area under the curve would be 4 x 50 = 200. But what does this 200 mean? It is the total distance traveled. For example, if you are going 50 MPH for 4 hours, then you have traveled 200 miles. The shaded area in the next graph represents the area under the curve from t = 0 to t = 4 hours.

This is very interesting! You have found that the velocity graph can tell you the distance traveled just by finding the area under the curve! But why would you use this graph to find distance? It's so much work, maybe not with this graph, but what if the velocity was not constant? What if the function was a "real" curve? You know that finding the area under a curve requires that you use Riemann Sums, and this isn't so easy sometimes without a computer. Why don't you just use the Distance Function?

The question becomes though, how do you get the Distance Function from the Velocity Function? In Chapter 2 you determined that the derivative of the Distance Function gives you the Velocity Function, wouldn't doing the reverse give you the Distance Function? If you take the anti-derivative of the Velocity Function you should get the Distance function. Below this is done both mathematically and graphically. The graph contains both functions.

Note: Notice how the y-axis is labeled D or V. This is to remind you that the y-values have different units depending on which curve you are looking at.

Now, if you need the distance all you need do is go to the desired time (the 4-hour mark in the example) and read off the graph (y-axis) the distance traveled, 200 miles.

You have just used the Fundamental Theorem of Calculus!! To find the area under the curve, all you have to do is evaluate the anti-derivative at the value 4. In the example, the anti-derivative of the Velocity Function, 50 is equal to 50 t. The same formula you used when you found the area!

You may wonder; what happens if you are looking for a different interval? Say from hour #2 to hour #4? It's just as easy! Just evaluate the anti-derivative evaluated at 4 and subtract the anti-derivative evaluated at 2. The next graph shows this.

The first shows the answer by finding the area under the curve (a Riemann Sum). The second uses the Fundamental Theorem.

The area under the Velocity curve this time is 50 times the difference between 2 and 4, or 2 x 50 = 100 miles traveled.

Using the anti-derivative method you would first evaluate at 4 ( 4 x 50 = 200 ) and then evaluate at 2 (2 x 50 = 100). Then you would subtract to obtain the answer; 200 - 100 = 100.

The next graph has been re-scaled so you can see both plots.

5.20 Fundamental Theorem of Calculus

To really make sure this theorem works you will use a more complicated function. The steps will be as follows:


1) Graph the function to see what it looks like making sure it is a positive one.

2) Find the area under the curve between 2 and 4 by using Riemann Sums (Trap Method).

3) Verify this by taking the Definite Integral.

4) Solve the Anti-derivative.

5) Evaluate using LiveMath's Evaluation-At Op, make a connection with the Definite Integral.