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

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

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.

1 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 Evaluate a definite integral using properties of definite integrals. 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 4.2. Example A Partition with Subintervals of Unequal Widths Riemann Sums Consider the region bounded by the graph of the x axis for 0 x 1, as shown and 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). 1

Definite Integrals Definite Integrals Basically, as we divide a region into an infinite number of

This is called the definite integral and is denoted by: where a and b are upper and lower limits.

Definite Integrals as Area Definite Integrals as Area As an example of Theorem 4.

2 Definite Integrals 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. Basically, if a function is continuous, then you can integrate it, (technically). Definite Integrals as Area Definite Integrals as Area As an example of Theorem 4.5, consider the region bounded by the graph of f(x) = 4x x 2 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 2

Definite Integrals Because the definite integral in the example below is negative, it does not represent the area of the region shown. Definite integrals can be positive, negative, or zero.

3 Definite Integrals Because the definite integral in the example below is negative, it does not represent the area of the region shown. 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]. Definite Integrals You can evaluate a definite integral in more than one way: You can use the limit definition & Reimann Sums You can check to see whether the definite integral represents the area of a common geometric region such as a rectangle, triangle, or semicircle. Example Areas of Common Geometric Figures Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula. a. Properties of Definite Integrals The definition of the definite integral of f on the interval [a, b] specifies that a < b. It is, however, convenient to extend the definition to cover cases in which a = b or a > b. Geometrically, the following two definitions seem reasonable. b. c. 3

Example Evaluating Definite Integrals a.

limits of integration are equal, you can write b.

larger region can be divided at x = c into two sub regions.

4 Example 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 Example Evaluating Definite Integrals The larger region can be divided at x = c into two sub regions. It follows that the area of the larger region is equal to the sum of the areas of the two smaller regions. cont d so what is Example Using the Additive Interval Property Properties of Definite Integrals Note that Property 2 of Theorem 4.7 can be extended to cover any finite number of functions. For example, 4

5 Example Evaluation of a Definite Integral Properties of Definite Integrals Evaluate using each of the following values. 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. Properties of Definite Integrals 4.4 Objectives The Fundamental Theorem of Calculus 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. 5

The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus We can now evaluate a definite integral using Riemann Sums (or the trapezoidal rule), & we can use geometric formulas, but what

6 The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus We can now evaluate a definite integral using Riemann Sums (or the trapezoidal rule), & we can use geometric formulas, but what are the problems with using these two methods? The two major branches of calculus are 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. The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus GUIDELINES FOR USING THE FUNDAMENTAL THEOREM OF CALCULUS * Provided you can find an antiderivative of f, you now have a way to evaluate a definite integral without having to use the limit of a sum. * * Example: 6

7 Why don't we need the "+C" any more? Example Evaluating a Definite Integral Evaluate each definite integral. Example Remember these problems from Section 4.3? We used geometry to find the areas let's do them now using the fundamental theorem of calculus: a. Definite Integrals Consider the region bounded by the graph of f(x) = 4x x 2 and the x axis, as shown. Find the area of the shaded region. b. 7

Example Finding the Average Value of a Function Find the average value of f(x) = 3x 2 2x on the interval [1, 4].

8 Average Value of a Function 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 of a Function Average value is like average height. (b a) is just the total width of the area we are integrating. Example Finding the Average Value of a Function Find the average value of f(x) = 3x 2 2x on the interval [1, 4]. 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. 8

9 The Second Fundamental Theorem of Calculus 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? The Second Fundamental Theorem of Calculus This result is generalized in the following theorem, called the Second Fundamental Theorem of Calculus. Example 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: 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. 9

1. 2. 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.

10 1. 2. 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. 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: Examples:

Integration. Copyright Cengage Learning. All rights reserved.

Integration. Copyright Cengage Learning. All rights reserved. 4 Integration Copyright Cengage Learning. All rights reserved. 1 4.3 Riemann Sums and Definite Integrals Copyright Cengage Learning. All rights reserved. 2 Objectives Understand the definition of a Riemann