Definition: the definite integral of f from a to b is provided that this limit exists. If it does exist, we say that is f integrable on [a,b] Sec 5.2:

Slides:

Advertisements

Similar presentations

5.2 Definite Integrals In this section we move beyond finite sums to see what happens in the limit, as the terms become infinitely small and their number.

Advertisements

Riemann sums, the definite integral, integral as area

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.

INTEGRALS 5. INTEGRALS We saw in Section 5.1 that a limit of the form arises when we compute an area.  We also saw that it arises when we try to find.

Chapter 5 Integrals 5.2 The Definite Integral In this handout: Riemann sum Definition of a definite integral Properties of the definite integral.

Copyright © Cengage Learning. All rights reserved. 5 Integrals.

5.2 Definite Integrals Quick Review Quick Review Solutions.

Riemann Sums and the Definite Integral Lesson 5.3.

Definite Integrals Sec When we find the area under a curve by adding rectangles, the answer is called a Rieman sum. subinterval partition The width.

Georg Friedrich Bernhard Riemann

Section 7.2a Area between curves.

6.3 Definite Integrals and the Fundamental Theorem.

Introduction to integrals Integral, like limit and derivative, is another important concept in calculus Integral is the inverse of differentiation in some.

Section 4.3 – Riemann Sums and Definite Integrals

5.2 Definite Integrals.

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.

Section 15.3 Area and Definite Integral

Section 5.2: Definite Integrals

1 §12.4 The Definite Integral The student will learn about the area under a curve defining the definite integral.

Learning Objectives for Section 13.4 The Definite Integral

Announcements Topics: -sections 7.3 (definite integrals) and 7.4 (FTC) * Read these sections and study solved examples in your textbook! Work On: -Practice.

ESSENTIAL CALCULUS CH04 Integrals. In this Chapter: 4.1 Areas and Distances 4.2 The Definite Integral 4.3 Evaluating Definite Integrals 4.4 The Fundamental.

CHAPTER Continuity The Definite Integral animation  i=1 n f (x i * )  x f (x) xx Riemann Sum xi*xi* xixi x i+1.

Introduction to Integration

5.6 Definite Integrals Greg Kelly, Hanford High School, Richland, Washington.

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

Warm Up 1) 2) 3)Approximate the area under the curve for 0 < t < 40, given by the data, above using a lower Reimann sum with 4 equal subintervals. 4)Approximate.

5.2 Definite Integrals. When we find the area under a curve by adding rectangles, the answer is called a Riemann sum. subinterval partition The width.

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.”

INTEGRALS We saw in Section 5.1 that a limit of the form arises when we compute an area. We also saw that it arises when we try to find the distance traveled.

Riemann Sums and The Definite Integral. time velocity After 4 seconds, the object has gone 12 feet. Consider an object moving at a constant rate of 3.

1. Does: ? 2. What is: ? Think about:. Finding Area between a Function & the x-axis Chapters 5.1 & 5.2 January 25, 2007.

Chapter 6 Integration Section 4 The Definite Integral.

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

Riemann Sums and Definite Integration y = 6 y = x ex: Estimate the area under the curve y = x from x = 0 to 3 using 3 subintervals and right endpoints,

Lesson 5-2 The Definite Integral. Ice Breaker See handout questions 1 and 2.

Chapter 6 Integration Section 5 The Fundamental Theorem of Calculus (Day 1)

4.3 Riemann Sums and Definite Integrals Definition of the Definite Integral If f is defined on the closed interval [a, b] and the limit of a Riemann sum.

Definite Integral df. f continuous function on [a,b]. Divide [a,b] into n equal subintervals of width Let be a sample point. Then the definite integral.

Riemann Sum. When we find the area under a curve by adding rectangles, the answer is called a Rieman sum. subinterval partition The width of a rectangle.

Area/Sigma Notation Objective: To define area for plane regions with curvilinear boundaries. To use Sigma Notation to find areas.

Definite Integrals & Riemann Sums

Announcements Topics: -sections 7.3 (definite integrals) and 7.4 (FTC) * Read these sections and study solved examples in your textbook! Work On: -Practice.

Definite Integrals. Definite Integral is known as a definite integral. It is evaluated using the following formula Otherwise known as the Fundamental.

Section 4.2 The Definite Integral. If f is a continuous function defined for a ≤ x ≤ b, we divide the interval [a, b] into n subintervals of equal width.

4.3 Riemann Sums and Definite Integrals

Announcements Topics: -sections 7.3 (definite integrals) and 7.4 (FTC) * Read these sections and study solved examples in your textbook! Work On: -Practice.

5.2 – The Definite Integral. Introduction Recall from the last section: Compute an area Try to find the distance traveled by an object.

1. Graph 2. Find the area between the above graph and the x-axis Find the area of each: 7.

The Definite Integral. Area below function in the interval. Divide [0,2] into 4 equal subintervals Left Rectangles.

5.2/3 Definite Integral and the Fundamental Theorem of Calculus Wed Jan 20 Do Now 1)Find the area under f(x) = 3 – x in the interval [0,3] using 3 leftendpt.

The Fundamental Theorem of Calculus Area and The Definite Integral OBJECTIVES  Evaluate a definite integral.  Find the area under a curve over a given.

Chapter 5 Integrals 5.2 The Definite Integral

Chapter 5 AP Calculus BC.

5.2 Definite Integral Tues Nov 15

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

Sec 5.2: The Definite Integral

Area & Riemann Sums Chapter 5.1

Advanced Mathematics D

Section 5.2 Definite Integrals.

Advanced Mathematics D

Copyright © Cengage Learning. All rights reserved.

Definition: Sec 5.2: THE DEFINITE INTEGRAL

AP Calculus December 1, 2016 Mrs. Agnew

Presentation transcript:

Definition: the definite integral of f from a to b is provided that this limit exists. If it does exist, we say that is f integrable on [a,b] Sec 5.2: THE DEFINITE INTEGRAL

Note 1: Sec 5.2: THE DEFINITE INTEGRAL Integral sign limits of integration lower limit a upper limit b integrand The procedure of calculating an integral is called integration. The dx simply indicates that the independent variable is x.

Note 2: Sec 5.2: THE DEFINITE INTEGRAL x is a dummy variable. We could use any variable

Note 3: Sec 5.2: THE DEFINITE INTEGRAL Riemann sum Riemann sum is the sum of areas of rectangles.

Note 4: Sec 5.2: THE DEFINITE INTEGRAL Riemann sum is the sum of areas of rectangles. area under the curve

Note 5: Sec 5.2: THE DEFINITE INTEGRAL If takes on both positive and negative values, the Riemann sum is the sum of the areas of the rectangles that lie above the -axis and the negatives of the areas of the rectangles that lie below the -axis (the areas of the gold rectangles minus the areas of the blue rectangles). A definite integral can be interpreted as a net area, that is, a difference of areas: where is the area of the region above the x-axis and below the graph of f, and is the area of the region below the x-axis and above the graph of f.

Note 6: Sec 5.2: THE DEFINITE INTEGRAL not all functions are integrable f(x) is cont [a,b] f(x) has only finite number of removable discontinuities f(x) has only finite number of jump discontinuities

Sec 5.2: THE DEFINITE INTEGRAL f(x) is cont [a,b] f(x) has only finite number of removable discontinuities f(x) has only finite number of jump discontinuities

Note 7: Sec 5.2: THE DEFINITE INTEGRAL the limit in Definition 2 exists and gives the same value no matter how we choose the sample points

Sec 5.2: THE DEFINITE INTEGRAL

Term-092

Example: Sec 5.2: THE DEFINITE INTEGRAL (a)Evaluate the Riemann sum for taking the sample points to be right endpoints and a =0, b =3, and n = 6. (b) Evaluate

Example: Sec 5.2: THE DEFINITE INTEGRAL (a)Set up an expression for as a limit of sums Example: Evaluate the following integrals by interpreting each in terms of areas.

Sec 5.2: THE DEFINITE INTEGRAL

Midpoint Rule Sec 5.2: THE DEFINITE INTEGRAL We often choose the sample point to be the right endpoint of the i-th subinterval because it is convenient for computing the limit. But if the purpose is to find an approximation to an integral, it is usually better to choose to be the midpoint of the interval, which we denote by.

Sec 5.2: THE DEFINITE INTEGRAL

Property (1) Sec 5.2: THE DEFINITE INTEGRAL Example:

Sec 5.2: THE DEFINITE INTEGRAL Property (2)

Sec 5.2: THE DEFINITE INTEGRAL Property (3)

Example: Sec 5.2: THE DEFINITE INTEGRAL Note: Property 1 says that the integral of a constant function is the constant times the length of the interval. Use the properties of integrals to evaluate

Sec 5.2: THE DEFINITE INTEGRAL Term-091

Sec 5.2: THE DEFINITE INTEGRAL

Term-092

Example: Sec 5.2: THE DEFINITE INTEGRAL Use Property 8 to estimate

Sec 5.2: THE DEFINITE INTEGRAL SYMMETRY Suppose f is continuous on [-a, a] and even Suppose f is continuous on [-a, a] and odd

Sec 5.2: THE DEFINITE INTEGRAL Term-102

Sec 5.2: THE DEFINITE INTEGRAL Term-102

Sec 5.2: THE DEFINITE INTEGRAL Term-091

Sec 5.2: THE DEFINITE INTEGRAL Term-091

Sec 5.2: THE DEFINITE INTEGRAL

Term-102

Sec 5.2: THE DEFINITE INTEGRAL Term-082

Sec 5.2: THE DEFINITE INTEGRAL Term-082

Sec 5.2: THE DEFINITE INTEGRAL Term-103

Sec 5.2: THE DEFINITE INTEGRAL Term-103

Sec 5.2: THE DEFINITE INTEGRAL Term-103

Sec 5.2: THE DEFINITE INTEGRAL Term-103

Sec 5.2: THE DEFINITE INTEGRAL Term-092

Sec 5.2: THE DEFINITE INTEGRAL Term-082