The Fundamental Theorem of Calculus Lesson 7.4. 2 Definite Integral Recall that the definite integral was defined as But … finding the limit is not often.

Slides:

Advertisements

Similar presentations

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

Advertisements

6 Integration Antiderivatives and the Rules of Integration

Copyright © Cengage Learning. All rights reserved. 13 The Integral.

Areas and Definite Integrals. Objectives Students will be able to Calculate a definite integral. Calculate the area between a curve and the x-axis over.

Homework Homework Assignment #2 Read Section 5.3

The Area Between Two Curves

The Area Between Two Curves Lesson When f(x) < 0 Consider taking the definite integral for the function shown below. The integral gives a negative.

The FTC Part 2, Total Change/Area & U-Sub. Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty.

The Fundamental Theorem of Calculus Inverse Operations.

Formal Definition of Antiderivative and Indefinite Integral Lesson 5-3.

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,

4-3 DEFINITE INTEGRALS MS. BATTAGLIA – AP CALCULUS.

Section 5.3 – The Definite Integral

Section 5.3: Evaluating Definite Integrals Practice HW from Stewart Textbook (not to hand in) p. 374 # 1-27 odd, odd.

5.c – The Fundamental Theorem of Calculus and Definite Integrals.

AP Calculus Definite Integrals Review (sections )

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.

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

4-4: The Fundamental Theorems Definition: If f is continuous on [ a,b ] and F is an antiderivative of f on [ a,b ], then: The Fundamental Theorem:

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 4.4 The Fundamental Theorem of Calculus Part II – The Second Fundamental Theorem.

The Fundamental Theorems of Calculus Lesson 5.4. First Fundamental Theorem of Calculus Given f is  continuous on interval [a, b]  F is any function.

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

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

F UNDAMENTAL T HEOREM OF CALCULUS 4-B. Fundamental Theorem of Calculus If f(x) is continuous at every point [a, b] And F(x) is the antiderivative of f(x)

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

SPECIALIST MATHS Calculus Week 6 Definite Integrals & Areas.

4-3: Riemann Sums & Definite Integrals

Antiderivatives Lesson 7.1A. Think About It Suppose this is the graph of the derivative of a function What do we know about the original function? Critical.

Integrals Related to Inverse Trig, Inverse Hyperbolic Functions

The Indefinite Integral

FTC Review; The Method of Substitution

The Fundamental Theorem of Calculus

Chapter 5 Integration. Indefinite Integral or Antiderivative.

5.a – Antiderivatives and The Indefinite Integral.

Substitution Lesson 7.2. Review Recall the chain rule for derivatives We can use the concept in reverse To find the antiderivatives or integrals of complicated.

Integration/Antiderivative. First let’s talk about what the integral means! Can you list some interpretations of the definite integral?

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.

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

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.

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

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

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

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc. Integration 5 Antiderivatives Substitution Area Definite Integrals Applications.

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.

INTEGRALS 5. If u = g(x), Summary If f is even, f(-x) = f(x), If f is odd, f(-x) = -f(x),

Announcements Topics: -sections 7.3 (definite integrals), 7.4 (FTC), and 7.5 (additional techniques of integration) * Read these sections and study solved.

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

(MTH 250) Lecture 19 Calculus. Previous Lecture’s Summary Definite integrals Fundamental theorem of calculus Mean value theorem for integrals Fundamental.

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

THE FUNDAMENTAL THEOREM OF CALCULUS Section 4.4. THE FUNDAMENTAL THEOREM OF CALCULUS Informally, the theorem states that differentiation and definite.

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

6 Integration Antiderivatives and the Rules of Integration

Definite Integration Say good-bye to C

4.4 The Fundamental Theorem of Calculus

Chapter 5 Integrals.

The Fundamental Theorems of Calculus

Section 5.4 Theorems About Definite Integrals

Fundamental Theorem of Calculus Indefinite Integrals

Chapter 4 Integration.

Antiderivatives Lesson 7.1A.

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus

Properties of Definite Integrals

Chapter 7 Integration.

The Indefinite Integral

The Fundamental Theorems of Calculus

5.1 Integrals Rita Korsunsky.

Presentation transcript:

The Fundamental Theorem of Calculus Lesson 7.4

2 Definite Integral Recall that the definite integral was defined as But … finding the limit is not often convenient We need a better way!

3 Fundamental Theorem of Calculus Given function f(x), continuous on [a, b] Let F(x) be any antiderivative of f Then we claim that The definite integral is equal to the difference of the two antiderivatives Shazzam !

4 What About + C ? The constant C was needed for the indefinite integral It is not needed for the definite integral The C's cancel out by subtraction

5 You Gotta Try It Consider What is F(x), the antiderivative? Evaluate F(5) – F(0)

6 Properties Bringing out a constant factor The integral of a sum is the sum of the integrals

7 Properties Splitting an integral f must be continuous on interval containing a, b, and c

8 Example Consider Which property is being used to find F(x), the antiderivative? Evaluate F(4) – F(0)

9 Try Another What is Hint … combine to get a single power of x What is F(x)? What is F(3) – F(1)?

10 When Substitution Is Used Consider u = 4m du = 12m 2 It is best to change the new limits to be in terms of u When m = 0, u = 2 When m = 3, u = 110

11 Area Why would this definite integral give a negative area? The f(x) values are negative You must take this into account if you want the area between the axis and the curve

12 Assignment Lesson 7.4A Page 399 Exercises 1 – 43 odd Lesson 7.4B Page 401 Exercises 53 – 65 odd