Section 4.3 – Riemann Sums and Definite Integrals

Slides:

Advertisements

Similar presentations

Riemann sums, the definite integral, integral as area

Advertisements

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.

Applying the well known formula:

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.

5.2 Definite Integrals Quick Review Quick Review Solutions.

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:

Riemann Sums & Definite Integrals Section 5.3. Finding Area with Riemann Sums For convenience, the area of a partition is often divided into subintervals.

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.

4.2 Area Under a Curve.

Georg Friedrich Bernhard Riemann

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]

Section 7.2a Area between curves.

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

5.2 Definite Integrals. Subintervals are often denoted by  x because they represent the change in x …but you all know this at least from chemistry class,

CHAPTER 4 SECTION 4.2 AREA.

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 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

CHAPTER 4 SECTION 4.3 RIEMANN SUMS AND DEFINITE INTEGRALS.

Chapter 5: The Definite Integral Section 5.2: Definite Integrals

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.

Introduction to Integration

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

Antidifferentiation: The Indefinite Intergral Chapter Five.

5.2 Definite Integrals Bernhard Reimann

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.”

Chapter 6 Integration Section 4 The Definite Integral.

5.2 Definite Integrals Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts.

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

X01234 f(x)f(x) AP Calculus Unit 5 Day 5 Integral Definition and intro to FTC.

Lesson 5-2R Riemann Sums. Objectives Understand Riemann Sums.

Riemann sums & definite integrals (4.3) January 28th, 2015.

5.2 Riemann Sums and Area. I. Riemann Sums A.) Let f (x) be defined on [a, b]. Partition [a, b] by choosing These partition [a, b] into n parts of length.

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.

Recall that R N, L N, and M N use rectangles of equal width Δx, whose heights are the values of f (x) at the endpoints or midpoints of the subintervals.

5.3 Definite Integrals. Example: Find the area under the curve from x = 1 to x = 2. The best we can do as of now is approximate with rectangles.

Definite Integrals & Riemann Sums

4-3: Riemann Sums & Definite Integrals Objectives: Understand the connection between a Riemann Sum and a definite integral Learn properties of definite.

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.

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

30. Section 5.2 The Definite Integral Table of Contents.

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

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.

Riemann Sums & Definite Integrals

Chapter 5 Integrals 5.2 The Definite Integral

Chapter 5 AP Calculus BC.

Area and the Definite Integral

5.2 Definite Integral Tues Nov 15

Section 6. 3 Area and the Definite Integral Section 6

6-4 Day 1 Fundamental Theorem of Calculus

Area and the Definite Integral

The Area Question and the Integral

6.2 Definite Integrals.

Sec 5.2: The Definite Integral

Area & Riemann Sums Chapter 5.1

Applying the well known formula:

1. Reversing the limits changes the sign. 2.

6.2 Definite Integrals.

Definition: Sec 5.2: THE DEFINITE INTEGRAL

Riemann sums & definite integrals (4.3)

6-2 definite integrals.

Section 4 The Definite Integral

Presentation transcript:

Section 4.3 – Riemann Sums and Definite Integrals

Riemann Sums The rectangles need not have equal width, and the height may be any value of f(x) within the subinterval. 1. Partition (divide) [a,b] into N subintervals. 2. Find the length of each interval: 3. Find any point ci in the interval [xi,xi-1]. c1 c2 c3 ci cN =x0 a x1 x2 x3 xi =xN b 4. Construct every rectangle of height f(ci) and base Δxi. 4. Find the sum of the areas.

Riemann Sums The norm of P, denoted ││P││, is the maximum of the lengths Δxi. a b As ││P││ gets closer to 0, the sum of the areas of the rectangles is closer to the actual area under the curve

Riemann Sums The norm of P, denoted ││P││, is the maximum of the lengths Δxi. a b As ││P││ gets closer to 0, the sum of the areas of the rectangles is closer to the actual area under the curve

Upper limit of integration Lower limit of integration Definite Integral The definite integral of f(x) over [a,b], denoted by the integral sign, is the limit of Riemann sums: Where the limit exists, we say that f(x) is integrable over [a,b]. Upper limit of integration Lower limit of integration

Notation Examples The definite integral that represents the area is… EX1: f(x) S a b Ex2: The area under the parabola y=x2 from 0 to 1

Theorem: The Existence of Definite Integrals If f(x) is continuous on [a,b], or if f(x) is continuous with at most finitely many jump discontinuities (one sided limits are finite but not equal), then f(x) is integrable over [a,b]. a b

Negative Area or “Signed” Area If a function is less than zero for an interval, the region between the graph and the x-axis represents negative area. Positive Area Negative Area

Definite Integral: Area Under a Curve If y=f(x) is integrable over a closed interval [a,b], then the area under the curve y=f(x) from a to b is the integral of f from a to b. Upper limit of integration Lower limit of integration

Example 1 Calculate .

Example 2 Calculate .

Rules for Definite Integrals Let f and g be functions and x a variable; a, b, c, and k be constant. Constant Constant Multiple Sum Rule Reversing the Limits Additivity

Constant Multiple Rule Given and Constant Rule Example 1 If , calculate . Sum Rule Constant Multiple Rule Given and Constant Rule

Example 2 If , calculate . Additivity Rule Given