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3 Objectives Understand the definition of a Riemann sum. Evaluate a definite integral using limits. Evaluate a definite integral using properties of definite integrals. 3

4 Riemann Sums In mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It may also be used to define the integration operation. The method was named after German mathematician Bernhard Riemann. Some examples are Upper Sums, Lower Sums, and Midpoint Sums like we learned about in Section

Example 1 A Partition with Subintervals of Unequal Widths Consider the region bounded by the graph of x 1, as shown in Figure 4.17.

5 Example 1 A Partition with Subintervals of Unequal Widths Consider the region bounded by the graph of x 1, as shown in Figure and the x axis for 0 Notice that the rectangles are not the same width. You don t have to have equal widths to do a Riemann Sum, (but it is easier to do if the subintervals have equal widths). Figure

6 Riemann Sums 6

7 Definite Integrals 7

8 Definite Integrals Basically, as we divide a region into an infinite number of rectangles, each having a width of, we get infinitely close to the actual area of the region. This is called the definite integral and is denoted by where a and b are upper and lower limits. 8

9 Definite Integrals 9

10 Definite Integrals Figure

11 Definite Integrals As an example of Theorem 4.5, consider the region bounded by the graph of f(x) = 4x x2 and the x axis, as shown in Figure Because f is continuous and nonnegative on the closed interval [0, 4], the area of the region is Figure

12 Definite Integrals You can evaluate a definite integral in two ways you can use the limit definition or you can check to see whether the definite integral represents the area of a common geometric region such as a rectangle, triangle, or semicircle. 12

13 Example 3 Areas of Common Geometric Figures Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula. a. b. (See next pages for solutions.) c. 13

14 Example 3(a) Solution This region is a rectangle of height 4 and width 2. Figure 4.23(a) 14

15 Example 3(b) Solution cont d This region is a trapezoid with an altitude of 3 and parallel bases of lengths 2 and 5. The formula for the area of a trapezoid is h(b1 + b2). Figure 4.23(b) 15

16 Example 3(c) Solution cont d This region is a semicircle of radius 2. The formula for the area of a semicircle is Figure 4.23(c) 16

Definite Integrals Because the definite integral in the example below is negative, it does not represent the area of the region shown in Figure 4.20.

17 Definite Integrals Because the definite integral in the example below is negative, it does not represent the area of the region shown in Figure Definite integrals can be positive, negative, or zero. For a definite integral to be interpreted as an area, the function f must be continuous and nonnegative on [a, b]. Figure

18 Properties of Definite Integrals 18

19 Properties of Definite Integrals The definition of the definite integral of f on the interval [a, b] specifies that a < b. Now, however, it is convenient to extend the definition to cover cases in which a = b or a > b. Geometrically, the following two definitions seem reasonable. For instance, it makes sense to define the area of a region of zero width and finite height to be 0. 19

20 Properties of Definite Integrals 20

21 Example 4 Evaluating Definite Integrals a. Because the sine function is defined at x = π, and the upper and lower limits of integration are equal, you can write b. The integral has a value of so you can write 21

22 Example 4 Evaluating Definite Integrals cont d In Figure 4.24, the larger region can be divided at x = c into two sub regions whose intersection is a line segment. Because the line segment has zero area, it follows that the area of the larger region is equal to the sum of the areas of the two smaller regions. Figure

23 Example 5 Using the Additive Interval Property 23

24 Properties of Definite Integrals Note that Property 2 of Theorem 4.7 can be extended to cover any finite number of functions. For example, 24

25 Example 6 Evaluation of a Definite Integral Evaluate using each of the following values. Solution: 25

26 Properties of Definite Integrals If f and g are continuous on the closed interval [a, b] and 0 f(x) g(x) for a x b, the following properties are true. First, the area of the region bounded by the graph of f and the x axis (between a and b) must be nonnegative. Second, this area must be less than or equal to the area of the region bounded by the graph of g and the x axis (between a and b ), as shown in Figure These two properties are generalized in Theorem 4.8. Figure

27 Properties of Definite Integrals 27

29 Objectives Evaluate a definite integral using the Fundamental Theorem of Calculus. Find the average value of a function over a closed interval. Understand and use the Second Fundamental Theorem of Calculus. 29

30 The Fundamental Theorem of Calculus 30

31 The Fundamental Theorem of Calculus The two major branches of calculus: differential calculus and integral calculus. At this point, these two problems might seem unrelated but there is a very close connection. The connection was discovered independently by Isaac Newton and Gottfried Leibniz and is stated in a theorem that is appropriately called the Fundamental Theorem of Calculus. 31

32 The Fundamental Theorem of Calculus 32

33 The Fundamental Theorem of Calculus The following guidelines can help you understand the use of the Fundamental Theorem of Calculus. We don't need "+C" any more because it just subtracts to zero! 33

34 Example 1 Solution 34

35 Example 3(a) Solution This region is a rectangle of height 4 and width 2. Figure 4.23(a) 35

36 Example 3(b) Solution cont d This region is a trapezoid with an altitude of 3 and parallel bases of lengths 2 and 5. The formula for the area of a trapezoid is h(b1 + b2). Figure 4.23(b) 36

37 Example 3(c) Solution cont d This region is a semicircle of radius 2. The formula for the area of a semicircle is Figure 4.23(c) 37

38 Average Value of a Function 38

39 Average Value of a Function In Figure 4.31 the area of the region under the graph of f is equal to the area of the rectangle whose height is the average value. Average value is like average height. Figure

40 Average Value of a Function b a is just the total width of the area we are integrating. 40

41 Example 4 Finding the Average Value of a Function Find the average value of f(x) = 3 x 2 2 x on the interval [1, 4]. Solution: The average value is given by Figure

42 The Second Fundamental Theorem of Calculus 42

43 The Second Fundamental Theorem of Calculus The definite integral of f on the interval [a, b] is defined using the constant b as the upper limit of integration and x as the variable of integration. A slightly different situation may arise in which the variable x is used in the upper limit of integration. To avoid the confusion of using x in two different ways, t is temporarily used as the variable of integration. 43

44 The Second Fundamental Theorem of Calculus 44

45 The Second Fundamental Theorem of Calculus If we are just told to integrate, we evaluate using the First Fundamental Theorem of Calculus: But what if we are doing the derivative of an integral. Then what would happen? 45

The Second Fundamental Theorem of Calculus This result is generalized in the following theorem, called the Second Fundamental Theorem of Calculus.

46 The Second Fundamental Theorem of Calculus This result is generalized in the following theorem, called the Second Fundamental Theorem of Calculus. Remember, this only works if you are taking the derivative of an integral, not the other way around, (integral of a derivative). Also, there must be a constant for the lower limit and x in the upper limit. 46

47 Example 7 Using the Second Fundamental Theorem of Calculus Evaluate Solution: Note that is continuous on the entire real line. So, using the Second Fundamental Theorem of Calculus, you can write 47

48 1. Examples:

The Second Fundamental Theorem of Calculus Remember we said there must be a constant for the lower limit and an x in the upper limit to use the Second Fundamental Theorem of Calculus.

49 The Second Fundamental Theorem of Calculus Remember we said there must be a constant for the lower limit and an x in the upper limit to use the Second Fundamental Theorem of Calculus. It turns out that you can also use the theorem when the lower limit is a constant and the upper limit is a function of x. The only difference is that we plug in the function of x for t (instead of just the x), and we also multiply by the derivative of the function we plugged in. Here is an example: 49

50 1. Examples:

4.3. Riemann Sums. Riemann Sums. Riemann Sums and Definite Integrals. Objectives

4.3. Riemann Sums. Riemann Sums. Riemann Sums and Definite Integrals. Objectives 4.3 Riemann Sums and Definite Integrals Objectives Understand the definition of a Riemann sum. Evaluate a definite integral using limits & Riemann Sums. Evaluate a definite integral using geometric formulas