5.3 Definite Integrals and Antiderivatives Greg Kelly, Hanford High School, Richland, Washington.

Slides:

Advertisements

Similar presentations

Mean Value Theorem for Derivatives4.2 Teddy Roosevelt National Park, North Dakota Photo by Vickie Kelly, 2002 Created by Greg Kelly, Hanford High School,

Advertisements

5.3 Definite Integrals and Antiderivatives Organ Pipe Cactus National Monument, Arizona Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie.

Mean Value Theorem for Derivatives4.2 Teddy Roosevelt National Park, North Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie.

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

5.4 Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998 Morro Rock, California.

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

Properties of Real Numbers

5.4 The Fundamental Theorem. The Fundamental Theorem of Calculus, Part 1 If f is continuous on, then the function has a derivative at every point in,

5.3 Definite Integrals and Antiderivatives Greg Kelly, Hanford High School, Richland, Washington.

Chapter 5 .3 Riemann Sums and Definite Integrals

5.3 Definite Integrals and Antiderivatives. 0 0.

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

4.4c 2nd Fundamental Theorem of Calculus. Second Fundamental Theorem: 1. Derivative of an integral.

Antiderivatives An antiderivative of f(x) is any function F(x) such that F’(x) = f(x)

First Fundamental Theorem. If you were being sent to a desert island and could take only one equation with you, might well be your choice. Here is a calculus.

Section 4.3 – Riemann Sums and Definite Integrals

Antiderivatives: Think “undoing” derivatives Since: We say is the “antiderivative of.

CHAPTER 4 SECTION 4.4 THE FUNDAMENTAL THEOREM OF CALCULUS.

4.4 The Fundamental Theorem of Calculus (Part 2) Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998 Morro Rock, California.

5.4 Fundamental Theorem of Calculus Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998 Morro Rock, California.

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

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

AP Calculus Mrs. Mongold. The Fundamental Theorem of Calculus, Part 1 If f is continuous on, then the function has a derivative at every point in, and.

Second Fundamental Theorem of Calculus

5.3 Definite Integrals and Antiderivatives. When I die, I want to go peacefully like my Grandfather did, in his sleep -- not screaming, like the passengers.

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

2.1: Rates of Change & Limits Greg Kelly, Hanford High School, Richland, Washington.

Mean Value Theorem for Derivatives4.2 Teddy Roosevelt National Park, North Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie.

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.

Essential Question: How is a definite integral related to area ?

Slide 5- 1 Quick Review. Slide 5- 2 Quick Review Solutions.

Chapter 5 Integrals 5.2 The Definite Integral

Indefinite Integrals or Antiderivatives

Mean Value Theorem for Derivatives

Mean Value Theorem 5.4.

2.3 The Product and Quotient Rules (Part 1)

Mean Value Theorem for Derivatives

4.4 The Fundamental Theorem of Calculus

Colorado National Monument

and Indefinite Integration (Part I)

Lesson 3: Definite Integrals and Antiderivatives

Mean Value Theorem for Derivatives

5.3 Inverse Function (part 2)

Properties of Definite Integrals

5.3 Definite Integrals and Antiderivatives

1. Reversing the limits changes the sign. 2.

5.4 First Fundamental Theorem

Mean Value Theorem for Derivatives

The Fundamental Theorem of Calculus

5.4 First Fundamental Theorem

2.2 Basic Differentiation Rules and Rates of Change (Part 1)

5.7 Inverse Trig Functions and Integration (part 2)

Trigonometric Substitutions

Mean Value Theorem and Antiderivatives

Colorado National Monument

Properties of Definite Integrals

2.4 The Chain Rule (Part 2) Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002.

2.2 Basic Differentiation Rules and Rates of Change (Part 1)

Mean Value Theorem for Derivatives

4.6 The Trapezoidal Rule Mt. Shasta, California

Mean Value Theorem for Derivatives

Mean Value Theorem for Derivatives

Finding constant of integration

5.4 Fundamental Theorem of Calculus

Fundamental Theorem of Calculus

Mean Value Theorem for Derivatives

Minimum and Maximum Values of Functions

5.3 Definite Integrals and Antiderivatives MLK JR Birthplace

Trigonometric Substitutions

5.3 Inverse Function (part 2)

Presentation transcript:

5.3 Definite Integrals and Antiderivatives Greg Kelly, Hanford High School, Richland, Washington

Page 269 gives rules for working with integrals, the most important of which are: 1. If the upper and lower limits are equal, then the integral is zero. 2. Reversing the limits changes the sign. 3. Constant multiples can be moved outside.

1. If the upper and lower limits are equal, then the integral is zero. 2. Reversing the limits changes the sign. 3. Constant multiples can be moved outside Integrals can be added and subtracted.

5-6. Integrals can be added and subtracted. 10. Intervals can be added (or subtracted.)

The average value of a function is the value that would give the same area if the function was a constant:

The mean value theorem for definite integrals says that for a continuous function, at some point on the interval the actual value will equal the average value. Average value of a function If f is continuous on then at some point c in, 

The math professor's six-year-old son knocks at the door of his father's study. "Daddy", he says. "I need help with a math problem I couldn't do at school." "Sure", the father says and smiles. "Just tell me what's bothering you." "Well, it's a really hard problem: There are four ducks swimming in a pond, when two more ducks come and join them. How many ducks are now swimming in the pond?" The professor stares at his son with disbelief: "You couldn't do that?! All you need to know is that = 6!" "Do you think, I'm stupid?! Of course, I know that = 6. But what does this have to do with ducks!?"