1. 6/3/2016Calculus - Santowski1 C.1.4 - The Fundamental Theorem of Calculus Calculus - Santowski

2. 6/3/2016Calculus - Santowski2 Lesson Objectives 1. Calculate simple definite integrals using the FTC 2. Calculate definite integrals using the properties of definite integrals 3. Determine total areas under curves using the FTC 4. Differentiate integral functions with variable (i) upper limits of integration and (ii) lower limits of integration 5. Apply definite integrals to a real world problems

3. 6/3/2016Calculus - Santowski3 (A) Review We have determined the area under curves in two ways to date: (i) estimating using a finite number of rectangles (using a variety of “rectangle construction methods” - LRAM, RRAM, MRAM, trapezoids) (ii) determining the exact area using an infinite number of rectangles and making use of various summation rules and limits at infinity (using the formula

4. 6/3/2016Calculus - Santowski4 (B) Review of Differentiation Recall that our study of integration followed the same development: (i) first we worked with average rates of change using secant lines (ii) we then made our estimates using secants more precise by having an infinitely small distance between the secant and tangency points and we developed a special formula

5. 6/3/2016Calculus - Santowski5 (B) Review of Differentiation Then in our development of calculus, we found that working with limits was tedious and could be replaced by a more efficient/convenient method ==> the derivative So our formula was “modified” as

6. 6/3/2016Calculus - Santowski6 (B) Review of Differentiation So our question now is this … does the same idea hold for area under curves …. Can our formula with limits be “modified” into something easier/more workable??

7. 6/3/2016Calculus - Santowski7 (C) Area under Curves - Investigation Let’s work with the function f(x) = x 2 and determine some areas under the curve Our lower bound will be a = 0 This time, we will change our upper limit so b will vary Again, we will use the TI-89 to help us with our calculations

8. 6/3/2016Calculus - Santowski8 (C) Area under Curves - Investigation Let’s start with b = 1, so our interpretation will be that we are trying to find the area under the curve f(x) = x 2 between x = 0 and x = 1 So then  x = (b - a)/n = (1 - 0)/n = 1/n And x i = a + i  x, so that f(x i ) = f(i/n) Then we will sum together an infinite number of rectangles ……

9. 6/3/2016Calculus - Santowski9 (C) Area under Curves - Investigation Now let’s make b = 2 So then  x = (b - a)/n = (2 - 0)/n = 2/n And x i = a + i  x, so that f(x i ) = f(2i/n) Then we will sum together an infinite number of rectangles ……

10. 6/3/2016Calculus - Santowski10 (C) Area under Curves - Investigation Now let’s make b = 3 So then  x = (b - a)/n = (3 - 0)/n = 3/n And x i = a + i  x, so that f(x i ) = f(3i/n) Then we will sum together an infinite number of rectangles ……

11. 6/3/2016Calculus - Santowski11 (C) Area under Curves - Investigation Now let’s make b = 4 So then  x = (b - a)/n = (4 - 0)/n = 4/n And x i = a + i  x, so that f(x i ) = f(4i/n) Then we will sum together an infinite number of rectangles ……

12. 6/3/2016Calculus - Santowski12 (C) Area under Curves - Investigation Now let’s make b = 5 So then  x = (b - a)/n = (5 - 0)/n = 5/n And x i = a + i  x, so that f(x i ) = f(5i/n) Then we will sum together an infinite number of rectangles ……

13. 6/3/2016Calculus - Santowski13 (C) Area under Curves - Investigation Now that we have seen a few calculations and some “data points”, let’s see what we really have ??? x012345 Area01/38/327/364/3125/3

14. 6/3/2016Calculus - Santowski14 (C) Area under Curves - Investigation Now let’s make b = x So then  x = (b - a)/n = (5 - 0)/n = x/n And x i = a + i  x, so that f(x i ) = f(xi/n) Then we will sum together an infinite number of rectangles ……

15. 6/3/2016Calculus - Santowski15 (D) Investigation: Conclusion So we started with the function f(x) = x 2 with which we were trying to find the area under the curve between any 2 points (let’s again use a & b) In working with the sums of infinite rectangles, we developed a “formula” that would allow us to quickly determine the area under the curve between a & b We saw that this Area Function was A(x) = 1/3x 3 Then, finally, what is the relationship between x 2 and 1/3x 3 ?? Our Area Function simply happens to be the antiderivative of the curve in question!!!!

16. 6/3/2016Calculus - Santowski16 (D) Investigation: Conclusion But we have introduced a new symbol for our antiderivative  the integral symbol (∫) So now we can put our relationships together Where is now referred to as a definite integral

17. 6/3/2016Calculus - Santowski17 (E) The Fundamental Theorem of Calculus The fundamental theorem of calculus shows the connection between antiderivatives and definite integrals Let f be a continuous function on the interval [a,b] and let F be any antiderivative of f. Then

18. 6/3/2016Calculus - Santowski18 (F) Examples Let’s work with the FTC on the following examples:

19. 6/3/2016Calculus - Santowski19 (G) Properties of Definite Integrals To further develop definite integrals and there applications, we need to establish some basic properties of definite integrals:

20. 6/3/2016Calculus - Santowski20 (H) Further Examples Let’s work with the FTC on the following examples

21. 6/3/2016Calculus - Santowski21 (H) Further Examples Interpret the definite integral in terms of areas.

22. 6/3/2016Calculus - Santowski22 (H) Further Examples So the answer to the question of interpreting the definite integral in terms of areas is: (i) 2.67 if the interpretation is as the difference between the positive area and the negative area (NET area) (ii) 12.77 if we interpret the answer to mean the TOTAL area of the regions

23. 6/3/2016Calculus - Santowski23 (H) Further Examples We are going to set up a convention here in that we would like the area to be interpreted as a positive number when we are working on TOTAL Area interpretations of the DI So, we will take the absolute value of negative “areas” or values Then the second consideration will be when a function has an x-intercept in the interval [a,b]  we will then break the area into 2 (or more) sub-intervals [a,c] & [c,b] and work with 2 separate definite integrals

24. 6/3/2016Calculus - Santowski24 (I) Homework Textbook, p424, Q3,5,6,36,38,40- 43,50,51,52 Handout from Calculus, 1988, Dunkley, p405-406, Q1,2 ANV, 3,4,5