Copyright © Cengage Learning. All rights reserved.

Slides:



Advertisements
Similar presentations
6.5 The Definite Integral In our definition of net signed area, we assumed that for each positive number n, the Interval [a, b] was subdivided into n subintervals.
Advertisements

Copyright © Cengage Learning. All rights reserved. 5 Integrals.
Riemann Sums & Definite Integrals Section 5.3. Finding Area with Riemann Sums For convenience, the area of a partition is often divided into subintervals.
Copyright © Cengage Learning. All rights reserved.
Aim: Riemann Sums & Definite Integrals Course: Calculus Do Now: Aim: What are Riemann Sums? Approximate the area under the curve y = 4 – x 2 for [-1, 1]
7 Applications of Integration
Integration Copyright © Cengage Learning. All rights reserved.
Integration 4 Copyright © Cengage Learning. All rights reserved.
6.3 Definite Integrals and the Fundamental Theorem.
Integration Copyright © Cengage Learning. All rights reserved.
If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by. As gets smaller, the approximation.
Copyright © Cengage Learning. All rights reserved The Area Problem.
Multiple Integration Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved. 4 Integrals.
Integration 4 Copyright © Cengage Learning. All rights reserved.
Chapter 6 INTEGRATION An overview of the area problem The indefinite integral Integration by substitution The definition of area as a limit; sigma notation.
1 Warm-Up a)Write the standard equation of a circle centered at the origin with a radius of 2. b)Write the equation for the top half of the circle. c)Write.
The Definite Integral Objective: Introduce the concept of a “Definite Integral.”
Copyright © Cengage Learning. All rights reserved.
4.3 Riemann Sums and Definite Integrals. Objectives Understand the definition of a Riemann sum. Evaluate a definite integral using limits. Evaluate a.
Integration 4 Copyright © Cengage Learning. All rights reserved.
Riemann sums & definite integrals (4.3) January 28th, 2015.
Applications of Integration 7 Copyright © Cengage Learning. All rights reserved.
4-3: Riemann Sums & Definite Integrals Objectives: Understand the connection between a Riemann Sum and a definite integral Learn properties of definite.
Definite Integrals, The Fundamental Theorem of Calculus Parts 1 and 2 And the Mean Value Theorem for Integrals.
4.3 Riemann Sums and Definite Integrals
Copyright © Cengage Learning. All rights reserved. 6 Applications of Integration.
Finite Sums, Limits, and Definite Integrals.  html html.
4 Integration.
Riemann Sums & Definite Integrals
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
7 Applications of Integration
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
7 Applications of Integration
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Section 6. 3 Area and the Definite Integral Section 6
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
MATH 1910 Chapter 4 Section 2 Area.
Copyright © Cengage Learning. All rights reserved.
Applications of Integration
7 Applications of Integration
Riemann Sums and Definite Integrals
RIEMANN SUMS AND DEFINITE INTEGRALS
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Numerical Integration
Copyright © Cengage Learning. All rights reserved.
6.2 Definite Integrals.
Objectives Approximate a definite integral using the Trapezoidal Rule.
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
7 Applications of Integration
Copyright © Cengage Learning. All rights reserved.
RIEMANN SUMS AND DEFINITE INTEGRALS
Riemann sums & definite integrals (4.3)
Applications of Integration
Presentation transcript:

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

4.3 Riemann Sums and Definite Integrals Copyright © Cengage Learning. All rights reserved.

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

Riemann Sums

Note the key new feature – unequal subinterval width

Riemann Sums The width of the largest subinterval of a partition  is the norm of the partition and is denoted by ||||. If every subinterval is of equal width, the partition is regular and the norm is denoted by For a general partition, the norm is related to the number of subintervals of [a, b] in the following way. So, the number of subintervals in a partition approaches infinity as the norm of the partition approaches 0. That is, ||||→0 implies that

Riemann Sums The converse of this statement is not true. For example, let n be the partition of the interval [0, 1] given by As shown in Figure 4.19, for any positive value of n, the norm of the partition n is So, letting n approach infinity does not force |||| to approach 0. In a regular partition, however, the statements ||||→0 and are equivalent. Figure 4.19

Definite Integrals

Example 2 – Evaluating a Definite Integral as a Limit Evaluate the definite integral Solution: The function f(x) = 2x is integrable on the interval [–2, 1] because it is continuous on [–2, 1]. Moreover, the definition of integrability implies that any partition whose norm approaches 0 can be used to determine the limit.

Example 2 – Solution cont’d For computational convenience, define  by subdividing [–2, 1] into n subintervals of equal width Choosing ci as the right endpoint of each subinterval produces

Example 2 cont’d So, the definite integral is given by

Definite Integrals Because the definite integral in Example 2 is negative, it does not represent the area of the region shown in Figure 4.20. 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], as stated in the following theorem. Figure 4.20

Definite Integrals Figure 4.21

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. c.

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

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)

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)

Properties of Definite Integrals

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. no area inside all dx < 0

Example 4 – Evaluating Definite Integrals cont’d In Figure 4.24, the larger region can be divided at x = c into two subregions 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 4.24

Example 5 – Using the Additive Interval Property

Properties of Definite Integrals Because the definite integral is defined as the limit of a sum => it inherits the properties of summation below: Note that Property 2 of Theorem 4.7 can be extended to cover any finite number of functions. For example,

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

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. (1) The area of the region bounded by the graph of f and the x-axis (between a and b) must be nonnegative. (2) 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 4.25.