4.2 and 4.6 filled in notes.notebook. December 08, Integration. Copyright Cengage Learning. All rights reserved.

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1 4 Integration Copyright Cengage Learning. All rights reserved. 1

2 4.2 Area Copyright Cengage Learning. All rights reserved. 2

3 Objectives Use sigma notation to write and evaluate a sum. Understand the concept of area. Approximate the area of a plane region. 3

4 Sigma Notation is used in figuring the area under a curve, so let's review it... 4

5 Sigma Notation This section begins by introducing a concise notation for sums. This notation is called sigma notation because it uses the uppercase Greek letter sigma, written as 5

6 Examples of Sigma Notation From parts (a) and (b), notice that the same sum can be represented in different ways using sigma notation. 6

7 Sigma Notation The following properties of summation can be derived using the associative and commutative properties of addition and the distributive property of addition over multiplication. (In the first property, k is a constant.) 7

8 Sigma Notation Examples = 8 8

9 Area 9

10 Area In Euclidean geometry, the simplest type of plane region is a rectangle. The definition for the area of a rectangle is A = bh. From this definition, you can develop formulas for the areas of many other plane regions. 10

11 Area For example, to determine the area of a triangle, you can form a rectangle whose area is twice that of the triangle, as shown in Figure 4.5. Figure

12 Area Once you know how to find the area of a triangle, you can determine the area of any polygon by subdividing the polygon into triangular regions, as shown in Figure 4.6. Figure

13 Area Finding the areas of regions other than polygons is more difficult. To find the area of a plane region, we will use the area of rectangles and the limiting process. 13

14 Upper and Lower Sums 14

15 Example Approximating the Area of a Plane Region Use five rectangles to find two approximations of the area of the region lying between the graph of f(x) = x and the x axis between x = 0 and x = 2. 15

16 Example Approximating the Area of a Plane Region Use five rectangles to find two approximations of the area of the region lying between the graph of f(x) = x and the x axis between x = 0 and x = 2. On the left there are 5 rectangles below the graph (lower sum). On the right, there are 5 rectangles above the graph, (upper sum). Figure

17 Example Lower Sum The lower sum is obtained by finding the area of each rectangle and adding them together. What is the width of each rectangle? How do we find it? What is the height of each rectangle? How do we find them? 17

18 Example Lower Sum The lower sum is obtained by finding the area of each rectangle and adding them together. The width of each rectangle is, and the height of each rectangle can be obtained by evaluating f at the right endpoint of each interval. 18

19 Example Lower Sum 19

20 Example Lower Sum Solution The sum of the areas of the five rectangles is Because each of the five rectangles lies inside the parabolic region, you can conclude that the area of the parabolic region is greater than

21 Example Upper Sum The upper sum is obtained by finding the area of each rectangle and adding them together. How do we find the width of each rectangle? How do we find the height of each rectangle? 21

22 Example Upper Sum The upper sum is obtained by finding the area of each rectangle and adding them together. The width of each rectangle is, and the height of each rectangle can be obtained by evaluating f at the left endpoint of each interval. 22

23 Example Upper Sum 23

24 Example Upper Sum Solution So, the sum is Because the parabolic region lies within the union of the five rectangular regions, you can conclude that the area of the parabolic region is less than By combining the results in parts (a) and (b), you can conclude that 6.48 < (Area of region) <

25 Upper and Lower Sums Consider a plane region bounded above by the graph of a nonnegative, continuous function y = f (x), as shown in Figure 4.9. The region is bounded below by the x axis, and the left and right boundaries of the region are the vertical lines x = a and x = b. Figure

26 Upper and Lower Sums To approximate the area of the region, begin by subdividing the interval [a, b] into n subintervals, each of width Δx = (b a)/n, as shown in Figure Figure

27 Upper and Lower Sums The sum of the areas of the inscribed rectangles is called a lower sum, and the sum of the areas of the circumscribed rectangles is called an upper sum. 27

28 Upper and Lower Sums From Figure 4.11, you can see that the lower sum s(n) is less than or equal to the upper sum S(n). Moreover, the actual area of the region lies between these two sums. Figure 4.11 Be careful for the lower (or upper) sum above, you do not always use the left side of the rectangle for the height. 28

29 Upper and Lower Sums This just means that as you increase the number of rectangles, you get a more accurate representation of the area. As n goes to infinity, the sum of the rectangle areas approach the actual area of the plane region. This is true whether you use an upper or a lower sum. 29

30 4.2 Lower & Upper Sum Example: Use four rectangles to find the lower and upper sums for the area of the region lying between the graph of f(x) = x 2 and the x axis between x = 0 and x = 2. 30

31 4.2 Midpoint Rule: Another way to estimate the sum of a plane region is to use the Mid point Rule. Here we use the middle of each rectangle to find the height instead of the left or right side like we did for the Upper or Lower Sums. An example is shown here: 31

32 4.2 Midpoint Rule: For example: The graph shown below is. Use four subintervals and the Midpoint Rule to find the area bounded by the equation, the x axis, x = 0, & x = 2. Each rectangle has a width of.5, but we use the middle of the rectangle for the height instead of the left or right side. The height of each rectangle going from left to right is f(.25), f(.75), f1.25), & f(1.75). So the estimated area is:.5[f(.25) + f(.75) + f(1.25) + f(1.75)] =

33 4.2 Midpoint Rule: Example: Use four subintervals to find the area of the region lying between the graph of f(x) = x 2 and the x axis between x = 0 and x = 2 using the Midpoint Rule. 33

34 4.6 Numerical Integration Copyright Cengage Learning. All rights reserved. 34

35 Objectives Approximate a definite integral using the Trapezoidal Rule. 35

36 The Trapezoidal Rule 36

37 The Trapezoidal Rule One way to approximate a definite integral is to use n trapezoids, as shown in Figure In the development of this method, assume that f is continuous and positive on the interval [a, b]. So, the definite integral represents the area of the region bounded by the graph of f and the x axis, from x = a to x = b. Figure

38 The Trapezoidal Rule First, partition the interval [a, b] into n subintervals, each of width x = (b a)/n, such that Then form a trapezoid for each subinterval (see Figure 4.43). This is the average of the bases multiplied by the height of the trapezoid. Figure

39 The Trapezoidal Rule 39

40 Example Approximation with the Trapezoidal Rule Use the Trapezoidal Rule to approximate Compare the results for n = 4 and n = 8, as shown in Figure Figure

41 Example When n = 4, x = π/4, then is approximately: 41

42 42

43 43

44 Example Solution When n = 4, x = π/4, and you obtain 44

45 Example Solution cont d When and you obtain 45