The Fundamental Theorem of Calculus

Slides:

Advertisements

Similar presentations

6 Integration Antiderivatives and the Rules of Integration

Advertisements

Riemann Sums and the Definite Integral Lesson 5.3.

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

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

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

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.

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:

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.

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

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.

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

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.

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

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),

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.

4.4 The Fundamental Theorem of Calculus

The Area Between Two Curves

Warm-Up Find the derivative.

6 Integration Antiderivatives and the Rules of Integration

Example, Page 321 Draw a graph of the signed area represented by the integral and compute it using geometry. Rogawski Calculus Copyright © 2008 W. H. Freeman.

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

Riemann Sums and the Definite Integral

Definite Integration Say good-bye to C

4.4 The Fundamental Theorem of Calculus

ISHIK UNIVERSITY FACULTY OF EDUCATION Mathematics Education Department

ISHIK UNIVERSITY FACULTY OF EDUCATION Mathematics Education Department

Unit 4:Mathematics Introduce various integration functions Aims

Chapter 5 Integrals.

The Mean Value Theorem for Integrals – Average Value

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. {image}

The Fundamental Theorems of Calculus

The Fundamental Theorem of Calculus Part 1 & 2

Integration Techniques

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. {image}

Unit 6 – Fundamentals of Calculus Section 6

Question from Test 1 Liquid drains into a tank at the rate 21e-3t units per minute. If the tank starts empty and can hold 6 units, at what time will it.

Section 5.4 Theorems About Definite Integrals

The Area Between Two Curves

Calculus for ENGR2130 Lesson 2 Anti-Derivative or Integration

5.3 – The Definite Integral and the Fundamental Theorem of Calculus

Fundamental Theorem of Calculus Indefinite Integrals

Lesson 3: Definite Integrals and Antiderivatives

Section 4.3 – Area and Definite Integrals

Integrals Related to Inverse Trig, Inverse Hyperbolic Functions

Chapter 4 Integration.

THE FUNDAMENTAL THEOREM OF CALCULUS

Antiderivatives Lesson 7.1A.

The Fundamental Theorem of Calculus

Integration: Evaluation and Applications

Warm Up 1. Find 2 6 2

Properties of Definite Integrals

Chapter 7 Integration.

The Indefinite Integral

The Fundamental Theorems of Calculus

AP Calculus December 1, 2016 Mrs. Agnew

Section 5.3 – The Definite Integral

5.1 Integrals Rita Korsunsky.

Section 5.3 – The Definite Integral

More Definite Integrals

Integration Techniques

The Fundamental Theorem of Calculus (4.4)

Integration by Substitution

(Finding area using integration)

Presentation transcript:

The Fundamental Theorem of Calculus Lesson 7.4

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

Why? Why is the area of the yellow rectangle at the end = a b

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 !

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

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

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

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

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

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

When Substitution Is Used Consider u = 4m3 + 2 du = 12m2 It is best to change the new limits to be in terms of u When m = 0, u = 2 When m = 3, u = 110

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

Assignment Lesson 7.4A Page 424 Exercises 1 – 43 odd Lesson 7.4B