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.

Slides:

Advertisements

Similar presentations

4.3 Riemann Sums and Definite Integrals

Advertisements

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

Riemann sums, the definite integral, integral as area

The Definite Integral. In the previous section, we approximated area using rectangles with specific widths. If we could fit thousands of “partitions”

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 Greg Kelly, Hanford High School, Richland, Washington.

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:

The Definite Integral.

THE DEFINITE INTEGRAL RECTANGULAR APPROXIMATION, RIEMANN SUM, AND INTEGRTION RULES.

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.

Chapter 5 .3 Riemann Sums and Definite Integrals

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]

First Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, Washington.

State Standard – 16.0a Students use definite integrals in problems involving area. Objective – To be able to use the 2 nd derivative test to find concavity.

Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

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,

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 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

Summation Notation Also called sigma notationAlso called sigma notation (sigma is a Greek letter Σ meaning “sum”) The series can be written.

1 5.2 – The Definite Integral. 2 Review Evaluate.

CHAPTER 4 SECTION 4.3 RIEMANN SUMS AND DEFINITE INTEGRALS.

Chapter 5: The Definite Integral Section 5.2: Definite Integrals

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

The Definite Integral.

Time velocity After 4 seconds, the object has gone 12 feet. Consider an object moving at a constant rate of 3 ft/sec. Since rate. time = distance: If we.

5.2 Definite Integrals Bernhard Reimann

Dr. Omar Al Jadaan Assistant Professor – Computer Science & Mathematics General Education Department Mathematics.

Sigma Notations Example This tells us to start with k=1 This tells us to end with k=100 This tells us to add. Formula.

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.

The Definite Integral Objective: Introduce the concept of a “Definite Integral.”

Calculus Date: 3/7/2014 ID Check Obj: SWBAT connect Differential and Integral Calculus Do Now: pg 307 #37 B #23 HW Requests: SM pg 156; pg 295 #11-17 odds,

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.

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.

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)

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

Riemann Sum. Formula Step 1 Step 2 Step 3 Riemann Sum.

4.3: Definite Integrals Learning Goals Express the area under a curve as a definite integral and as limit of Riemann sums Compute the exact area under.

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.

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.

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

Finite Sums, Limits, and Definite Integrals.  html html.

Chapter 5 Integrals 5.2 The Definite Integral

Riemann Sums and The Definite Integral

Chapter 5 AP Calculus BC.

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

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

Area and the Definite Integral

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

Area and the Definite Integral

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

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

Riemann Sums and Integrals

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

6.2 Definite Integrals.

Definition: Sec 5.2: THE DEFINITE INTEGRAL

Riemann sums & definite integrals (4.3)

6-2 definite integrals.

Presentation transcript:

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 is called a subinterval. The entire interval is called the partition. Subintervals do not all have to be the same size.

subinterval partition 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 for the area gets better. if P is a partition of the interval

is called the definite integral of over. If we use subintervals of equal length, then the length of a subinterval is: The definite integral is then given by:

Leibnitz introduced a simpler notation for the definite integral: Note that the very small change in x becomes dx.

Limit of Riemann Sum = Definite Integral

Integration Symbol lower limit of integration upper limit of integration integrand variable of integration (dummy variable) It is called a dummy variable because the answer does not depend on the variable chosen.

The Definite Integral

Existence of Definite Integrals All continuous functions are integrable.

Example Using the Notation

Area Under a Curve (as a Definite Integral) Note: A definite integral can be positive, negative or zero, but for a definite integral to be interpreted as an area the function MUST be continuous and nonnegative on [a, b].

Area

Integrals on a Calculator

Example Using NINT

Properties of Definite Integrals

Order of Integration

Zero

Constant Multiple

Sum and Difference

Additivity

1.If f is integrable and nonnegative on the closed interval [a, b], then 2.If f and g are integrable on the closed interval [a, b] and for every x in [a, b], then Preservation of Inequality