1. 5.2 Definite Integrals Greg Kelly, Hanford High School, Richland, Washington

2. Area and the Definite Integral How would we find the area under the following curve and above the x-axis? The work is a little more complex, but the approach is the same… area under the curve! f(x) x

3. We could use rectangles to approximate the area. First, divide the horizontal base with a partition. We will start with a partition that has two equal subintervals: [0,1] and [1,2] f(x) x Area and the Definite Integral

4. In this slide, the subinterval rectangles have been drawn using the right-hand endpoints of each subinterval. We could have used the left endpoints, the midpoints of the intervals, or any point in each subinterval. 0 1 2 1unit 2 2 +1 = 5 1 2 + 1 = 2 Area and the Definite Integral

5. 0 1 2 1 5 2 1 The area of the first rectangle is 2 and is greater than the area under the curve. The area of the second rectangle is 5 and is also greater than the area under the curve. The total area of the rectangles is 2 + 5 = 7 which is greater than the desired area under the curve. Area and the Definite Integral

6. In this slide, let’s draw the rectangles using the left-hand endpoints. 0 1 2 1unit 1 2 +1 = 2 0 2 + 1 = 1 Area and the Definite Integral

7. The area of the first rectangle is 1 x 1 = 1 and is less than the area under the curve. 0 1 2 1unit 2 1 The area of the second rectangle is 1 x 2 = 2 and is also less than the area under the curve. The total area of the rectangles using the left-hand endpoints is 1 + 2 = 3 which is less than the desired area under the curve. Now, let’s calculate the areas of these new rectangles. Area and the Definite Integral

8. Now we know that the area under the curve must be between 3 and 7. Can we get a more accurate answer? Yes! And to do so, we need to construct more subintervals using a finer partition. Area and the Definite Integral

9. Area Under A Curve The area under a curve on an interval can be approximated by summing the areas of individual rectangles on the interval. Area and the Definite Integral By using a finer partition (one with more subintervals), we make each rectangle narrower (increasing their number), and we get an area value that is closer to the true area under the curve. By taking the limit as n (the number of rectangles) approaches infinity, the actual area is approached.

10. Definition: A sum such as the one below is called a Riemann sum: Area Under A Curve Area and the Definite Integral

11. Let f(x) be a nonnegative, continuous function on the closed interval [a, b]. Then, the area of the region under the graph of f(x) is given by where x 1, x 2, x 3, … x n are arbitrary points in the n subintervals of [a,b] of equal width. Area Under A Curve The area under a curve can be approached by taking an infinite Riemann sum. Area and the Definite Integral

12. On the following six slides, you will see the number of rectangles increasing for the graph of y = x 3 on the interval [-1, 2]. Observe the way that the area of rectangles more closely approximates the area under the curve as the number of rectangles increases. Area and the Definite Integral

19. When there are 10 rectangles, Area 2.4675 20 rectangles, Area 3.091875 40 rectangles, Area 3.416719 100 rectangles, Area 3.615675 1000 rectangles, Area 3.736507 5000 rectangles, Area 3.7473 As you can see, the sum of the areas of the rectangles increases as you increase the number of rectangles. As the number of rectangles increases, they begin to better “fit” the curve giving a closer and closer approximation of the area (3.75) under the curve and above the x-axis. Area and the Definite Integral

20. We are almost done with this lesson in that we can now turn our attention to the definition and discussion of the definite integral for all functions, not just nonnegative functions. Area and the Definite Integral

21. Let f(x) be defined on [a,b]. If exists for all choices of representative points x 1, x 2, x 3, … x n in the n subintervals of [a,b] of equal width, then this limit is called the definite integral of f(x) from a to b and is denoted by. The number a is the lower limit of integration, and the number b is the upper limit of integration. Definition: The Definite Integral Area and the Definite Integral

22. Definition: The Definite Integral Thus As long as f(x) is a continuous function on a closed interval, it has a definite integral on that interval. f(x) is said to be integrable when its integral exists. Area and the Definite Integral

23. The definite integral of a function is a number. example: The indefinite integral of a function is another function. example: The Definite Integral Please make this important distinction between the indefinite integral of a function and the definite integral of a function: Area and the Definite Integral

24. If f(x) is a nonnegative, continuous function on [a, b], then is equal to the area of the region under the graph of f(x) on [a, b]. The Definite Integral Let’s take a quick look at the geometric interpretation of the definite integral, and we’ll be done. Area and the Definite Integral

25. If f(x) is a nonnegative, continuous function on [a, b], then is equal to the area of the region under the graph of f(x) on [a, b]. The Definite Integral f(x) y x a b What happens if f(x) is not always nonnegative? Area and the Definite Integral

26. If f(x) is simply a continuous function on [a, b], then is equal to the area of the region below the graph of f(x) and above the x-axis minus the area of the region above the graph of f(x) and below the x-axis on [a, b]. The Definite Integral f(x) y x a b Area of region 1 – Area of region 2 + Area of region 3 Area and the Definite Integral

27. Use the graph of the function to find the following definite integrals.

32. Area from x=0 to x=1 Example: Find the area under the curve from x = 1 to x = 2. Area from x=0 to x=2 Area under the curve from x = 1 to x = 2.

33. Example: Find the area between the x-axis and the curve from to. pos. neg.

34. Evaluate the integral

35. Find the area between the x-axis and the curve from -4 to 4. I II III 15.219+3.771+15.219=34.21

36. Use Integrals to find the shaded area I II III IV

37. Use Integrals to find the shaded area I II III IV 4 1/3

38. Use Integrals to find the shaded area I II III IV 4 1/3 10 24

39. Use Integrals to find the shaded area I II III IV 4 1/3 10 24 4 1/3

40. Use Integrals to find the shaded area I II III IV 4 1/3 10 24 4 1/3 Total Area= 4 1/3 + 24 + 10 + 4 1/3=42 2/3

41. P 267 7, 8, 13, 14 P 274 7-12, 17-20