1. Section 4.4 The Fundamental Theorem of Calculus

2. We have two MAJOR ideas to examine in this section of the text. We have been hinting for awhile, sometimes subtly and sometimes obviously, that there is an intimate connection between the calculation of areas and the antiderivative of a function.

3. Section 4.4 The Fundamental Theorem of Calculus  All of our summation formulas on page 260 share the feature that a power of i in the summation results in a power of n that is one degree higher. Here, we formally state the relationship with the Fundamental Theorem of Integral Calculus:

4. Section 4.4 The Fundamental Theorem of Calculus So, what does that formula mean? The notation that we are using here is that F(x) is an antiderivative of f (x). Let’s look at an example of this and we will check our result against a graphical representation. What would be the area enclosed by the line y = 2x – 3, the x -axis, and the lines x =1 and x = 4? So, the area can be computed by

5. Section 4.4 The Fundamental Theorem of Calculus Does a graph support this answer? Here is the graph I generated and there are two triangles to examine.

6. Section 4.4 The Fundamental Theorem of Calculus The first triangle has base of 0.5, height of 1 and is entirely below the x -axis, so this area is -0.25. The other triangle has a base of 2.5 and a height of 5. It is entirely above the x -axis, so its area is 6.25 for a total area of 6.25 – 0.25 = 6. This matches the integral calculation.

7. Section 4.4 The Fundamental Theorem of Calculus Let’s try one more example:  What is the area encompassed by the region whose boundaries are the x -axis, the lines x = 2 and x = 8, and the curve  The fundamental theorem of integral calculus tells us that this should be

8. Section 4.4 The Fundamental Theorem of Calculus Checking this calculation on our calculator, we see that the area is, in fact, correct. The second big idea is that of the mean value of a function on a domain. Just like the mean value theorem for derivatives tells us that there is a specific value in a region where the rate of change is equal to the average rate of change, the mean value theorem for integrals tells us that there is a place where f ( c ) is equal to the average value of the integral.

9. Section 4.4 The Fundamental Theorem of Calculus The mean value theorem for integrals leads us to this helpful definition presented on page 286. For some value of c in the interval [ a,b ] the following equation is true

10. Section 4.4 The Fundamental Theorem of Calculus So, what is the average value of the function on the interval [0, 5]? We solve the equation