The Definite Integral as an Accumulator Bob Arrigo Scarsdale High School Scarsdale, NY

Slides:

Advertisements

Similar presentations

{ Ch. 5 Review: Integrals AP Calculus. 5.2: The Differential dy 5.2: Linear Approximation 5.3: Indefinite Integrals 5.4: Riemann Sums (Definite Integrals)

Advertisements

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As noted in Theorem 1, the sign of the second derivative on an interval indicates the concavity.

5046: Modeling with the Definite Integral AP Calculus.

Integral as Accumulation What information is given to us by the area underneath a curve?

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

Getting the Most Out of the Fundamental Theorem Dan Kennedy Baylor School Chattanooga, TN

Homework Homework Assignment #1 Read Section 5.2 Page 308, Exercises: 1 – 25(EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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

5.2 Definite Integrals Quick Review Quick Review Solutions.

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

CALCULUS II Chapter 5. Definite Integral Example.

Riemann Sums and the Definite Integral Lesson 5.3.

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]

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

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

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

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.

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

SECTION 4-4 A Second Fundamental Theorem of Calculus.

Miss Battaglia AB/BC Calculus.  Connects differentiation and integration.  Integration & differentiation are inverse operations. If a function is continuous.

4009 Fundamental Theorem of Calculus (Part 2) BC CALCULUS.

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Sections 5.5 & 5.FT The Fundamental Theorems of.

1 5.2 – The Definite Integral. 2 Review Evaluate.

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.

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. MCS 122 Chapter 5 Review.

Calculus and Analytic Geometry I Cloud County Community College Fall, 2012 Instructor: Timothy L. Warkentin.

MAT 3751 Analysis II 5.2 The Riemann Integral Part I

Section 7.4: Arc Length. Arc Length The arch length s of the graph of f(x) over [a,b] is simply the length of the curve.

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

Functions Defined by Integrals

MAT 212 Brief Calculus Section 5.4 The Definite Integral.

1 5.d – Applications of Integrals. 2 Definite Integrals and Area The definite integral is related to the area bound by the function f(x), the x-axis,

5.1 Estimating with Finite Sums Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002 Greenfield Village, Michigan.

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.

Section 9.1 Arc Length. FINDING THE LENGTH OF A PLANE CURVE Divide the interval [a, b] into n equal subintervals. Find the length of the straight line.

4032 Fundamental Theorem AP Calculus. Where we have come. Calculus I: Rate of Change Function.

Fundamental Theorem AP Calculus. Where we have come. Calculus I: Rate of Change Function.

Section 4.3 Day 1 Riemann Sums and Definite Integrals AP Calculus BC.

Warm up Problems 1) Approximate using left- and right- hand Riemann sums, n = 6. 2) 3) Find without a calculator.

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.

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.

7.4 Arc Length. I. Arc Length A.) Given the curve f (x), we can approximate the length from x = a to x = b by using the distance formula.

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

Lecture 19 Approximating Integrals. Integral = “Area”

Section 4.3 Day 2 Riemann Sums & Definite Integrals AP Calculus BC.

5.3 Definite Integrals and Riemann Sums. I. Rules for Definite Integrals.

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.

Definite Integrals, The Fundamental Theorem of Calculus Parts 1 and 2 And the Mean Value Theorem for Integrals.

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.

Wednesday, April 6, 2016MAT 145 Please review TEST #3 Results and see me with questions!

[5-4] Riemann Sums and the Definition of Definite Integral Yiwei Gong Cathy Shin.

Chapter 5 AP Calculus BC.

Do Now - #22 and 24 on p.275 Graph the function over the interval. Then (a) integrate the function over the interval and (b) find the area of the region.

Riemann Sums as Estimates for Definite Integrals

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Riemann Sums and the Definite Integral

The Definite Integral as an Accumulator

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

Area and the Definite Integral

Definite Integrals Rizzi – Calc BC.

Riemann Sums as Estimates for Definite Integrals

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

Sr. lecturer in mathematics KHEMUNDI COLLEGE , DIGAPAHANDI.

Presentation transcript:

The Definite Integral as an Accumulator Bob Arrigo Scarsdale High School Scarsdale, NY

Traditional applications of the Definite Integral prior to the Calculus reform movement Area, volume, total distance traveled.. (AB) Arc length, work.. (BC) Mass, fluid pressure.. (Some college Calculus courses)

Calculus Reform in the early 90’s brought in “broader”, more robust applications of the definite integral…… most prominently, use of the definite integral to calculate “net change”, or “accumulated change.”

Types of Integrals Definite Integrals…limits of Riemann sums …”summing up infinitely many infinitesimally small products” Indefinite Integrals….a family of functions Integral functions….functions defined by an integral

The definite integral provides net change in a quantity over time. The definite integral of a rate function yields accumulated change of the associated function over some interval.

Motivate with a water flow problem: The rate at which water flows into a tank, in gallons per hour, is given by a differentiable function R of time t. Values of R are given at various times t during a 24 hour period. Approximate the number of gallons of water that flowed into the tank over the 24 hour period. tR(t)

This is an approximation for the total flow in gallons of water from the pipe in the 24-hour period.

Summing up lots of distances, each of which equals the product (rate)(time)

Method I to get the total distance traveled: Break up the interval [0,6] into smaller and smaller subintervals. To get the actual distance traveled, use more, smaller subintervals.

tv(t) Method II

Since method I and method II, both yield total distance, We get: Answer method I = Answer method II

Since method I and method II both yield total distance, We get: Answer method I = Answer method II

End Amt = Start Amt + NET CHANGE

Since is positive for and is negative for, the maximum value for occurs at time.