Integral Review Megan Bryant 4/28/2015. Bernhard Riemann  Bernhard Riemann (1826-1866) was an influential mathematician who studied under Gauss at the.

Slides:

Advertisements

Similar presentations

Follow the link to the slide. Then click on the figure to play the animation. A Figure Figure

Advertisements

5/16/2015 Perkins AP Calculus AB Day 5 Section 4.2.

Riemann Sums Jim Wang Mr. Brose Period 6. Approximate the Area under y = x² on [ 0,4 ] a)4 rectangles whose height is given using the left endpoint b)4.

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 4 THE DEFINITE INTEGRAL.

Definite Integrals Finding areas using the Fundamental Theorem of Calculus.

The First Fundamental Theorem of Calculus. First Fundamental Theorem Take the Antiderivative. Evaluate the Antiderivative at the Upper Bound. Evaluate.

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:

Wicomico High School Mrs. J. A. Austin AP Calculus 1 AB Third Marking Term.

The Integral chapter 5 The Indefinite Integral Substitution The Definite Integral As a Sum The Definite Integral As Area The Definite Integral: The Fundamental.

Georg Friedrich Bernhard Riemann

A REA A PPROXIMATION 4-B. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.

6.3 Definite Integrals and the Fundamental Theorem.

Homework questions thus far??? Section 4.10? 5.1? 5.2?

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.

7.4: The Fundamental Theorem of Calculus Objectives: To use the FTC to evaluate definite integrals To calculate total area under a curve using FTC and.

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,

5.2 Definite Integrals.

Areas & Definite Integrals TS: Explicitly assessing information and drawing conclusions.

1 Copyright © 2015, 2011 Pearson Education, Inc. Chapter 5 Integration.

Chapter 6 The Definite Integral. § 6.1 Antidifferentiation.

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

Fundamental Theorem of Calculus: Makes a connection between Indefinite Integrals (Antiderivatives) and Definite Integrals (“Area”) Historically, indefinite.

Maybe add in chad’s idea of the area of polygons limiting to the area of a circle. Nice animation to link to online At

5.4 Fundamental Theorem of Calculus Quick Review.

CHAPTER 4 SECTION 4.3 RIEMANN SUMS AND DEFINITE INTEGRALS.

Mathematics. Session Definite Integrals –1 Session Objectives  Fundamental Theorem of Integral Calculus  Evaluation of Definite Integrals by Substitution.

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

Problem of the Day If f ''(x) = x(x + 1)(x - 2) 2, then the graph of f has inflection points when x = a) -1 onlyb) 2 only c) -1 and 0 onlyd) -1 and 2.

Antidifferentiation: The Indefinite Intergral Chapter Five.

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

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

To find the area under the curve Warm-Up: Graph. Area under a curve for [0, 3]  The area between the x-axis and the function Warm-up What is the area.

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,

WS: Riemann Sums. TEST TOPICS: Area and Definite Integration Find area under a curve by the limit definition. Given a picture, set up an integral to calculate.

Johann Bernoulli 1667 – 1748 Johann Bernoulli 1667 – 1748 Johann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in.

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

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.

Properties of Integrals. Mika Seppälä: Properties of Integrals Basic Properties of Integrals Through this section we assume that all functions are continuous.

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.

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.

5-7: The 1 st Fundamental Theorem & Definite Integrals Objectives: Understand and apply the 1 st Fundamental Theorem ©2003 Roy L. Gover

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.

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

Warmup 11/30/15 How was your Thanksgiving? What are you most thankful for in your life? To define the definite integralpp 251: 3, 4, 5, 6, 9, 11 Objective.

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.

Application of the Integral

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

ISHIK UNIVERSITY FACULTY OF EDUCATION Mathematics Education Department

Riemann Sums Approximate area using rectangles

6.2 Definite Integrals.

5.3 – The Definite Integral and the Fundamental Theorem of Calculus

Accumulation AP Calculus AB Days 11-12

Sec 5.2: The Definite Integral

Applying the well known formula:

Riemann Sums and Integrals

Definition: Sec 5.2: THE DEFINITE INTEGRAL

76 – Riemann Sums – Rectangles Day 2 – Tables Calculator Required

AP Calculus December 1, 2016 Mrs. Agnew

Area Under a Curve Riemann Sums.

Calculus I (MAT 145) Dr. Day Wednesday April 17, 2019

Chapter 5 Integration.

Jim Wang Mr. Brose Period 6

Calculus I (MAT 145) Dr. Day Monday April 15, 2019

Presentation transcript:

Integral Review Megan Bryant 4/28/2015

Bernhard Riemann  Bernhard Riemann ( ) was an influential mathematician who studied under Gauss at the University of Gottingen.  He made major contributions to mathematics including complex variables, number theory, and the foundations of geometry.  We use Riemann’s definition of the definite integral.

Definite Integrals

Integration Terms  Integrand: f(x)  Limits of Integration: a,b  Upper Limit: b  Lower Limit: a

Riemann Sums

Left Riemann Sums  For a left Riemann sum, the endpoints that are used to construct your ‘rectangles’ are a, x 1,…,b-1, but do not include b (the right endpoint).  Left Riemann sums underestimate the area under the curve.

Right Riemann Sums  For a right Riemann sum, the endpoints that are used to construct your ‘rectangles’ are a+1, x 1,…,b, but do not include a (the left endpoint).  Right Riemann sums over estimate the area under the curve.

Mid Riemann Sums  For a mid Riemann sum, the endpoints that are used to construct your ‘rectangles’ are (a + x 1 )/2, (x 2 + x 3 )/2 …,(x b-1 +b)/2  Mid Riemann sums provide a balance between under (left) and over (right) estimating the area.

Left, Right, and Mid Points

Riemann Sum Example  Evaluate the Riemann sum for f(x) = x 3 -6x taking the sample points to be left endpoints and a = 0, b = 3, and n = 6.

Riemann Example: Find the Width

Riemann Example: Graph the Rectangles Remember, we are looking for the left Riemann sum with n=6 and a width of ½.

Riemann Sum: Final Area  To calculate the Riemann sum, find the area of all 6 rectangles and add them together.

Riemann Sum: Right  Now, let’s look at the right Riemann sum.  We will still have n = 6 and thus our width will remain the same.

Riemann Sums: Right Area Is this a better estimate? Why or why not? Is there a better way?

Mid Riemann Sum  Let’s try the Midpoint. We still have n = 6 and a width of ½.

Mid Riemann Sum Area This is a better estimate than the left and right Riemann sums separately. Midpoints often give a better estimate of the area under the curve.

First Fundamental Theorem of Calculus  The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function’s integral.  The first fundamental theorem of calculus is that the definite integration of a function is related to its anti derivative.

Example 1

Example 2