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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:
Example Evaluate the definite integral using a Riemann Sum the fundamental theorem:
Try This Evaluate the definite integral using the fundamental theorem:
Try This Evaluate the definite integral using the fundamental theorem: 4
Try Me Evaluate the definite integral using the fundamental theorem: 1
Example Find the area bounded by,the x axis, and the lines x =-2 and x =2 -2 2
Try This Find the area bounded by, the x axis, the y axis and the line x =2. 0 2 sq. units
Important Idea Provided you can find an antiderivative of f, you can now evaluate a definite integral without using the limit of a Riemann sum.
Try This Evaluate: =0
Assignment 1. 291/1-21odd
Definition Write this down… If f is continuous on [ a,b ], then there exists a number c in [ a,b ] such that Mean Value Theorem for Integrals:
Analysis f( c) c a b The area of the rectangle is the same as the area under the curve from a to b The area of the rectangle is f (c)(b-a)
Example Find the value c guaranteed by the Mean Value Theorem for Integrals for f(x) over the specified interval: for [1,3]
Warm-Up Find the value c guaranteed by the Mean Value Theorem for Integrals for f(x) over the specified interval: for [1,3]
Definition If f is integrable over [ a,b ], then the average value of f on the interval [ a,b ] is:
Analysis Average Value a b f(x) Average Value Since area is height times width, area divided by width equals height which is average value.
Analysis area divided by width =average value
Example Find the average value of f (x)=4-x 2 on [0,3]
Try me Find the average value of f(x)=x 3 on [1,3]
Important Idea A definite integral is normally a number, but…sometimes it is helpful to write a definite integral as a function.
Example What is different? The answer is a function and not a number.
Example Find the area under and above the x axis and between the vertical lines x =0 & x =1; x =0 & x =2, and x =0 & x =3
Try This Evaluate: For
Definition If f is continuous on an open interval containing a, then for every x in the interval: The 2 nd Fund. Theorem: This is a constant
Example Integrate to find F as a function of x and then demonstrate the second fundamental theorem by differentiating the result:
Important Idea The second fundamental theorem states that the derivative of the integral of a function is the function evaluated at the upper limit of integration.
Definition If f is continuous on an open interval containing a, then for every x in the interval: The 2 nd Fund. Theorem: Chain rule version
Try This Evaluate:
Try This Evaluate:
Assignment 1. 291/1-21odd 2. 35-47 odd, 53-59 odd, 81-91 odd (slides 15-31)
Lesson Close Name and explain three important theorems mentioned in this lesson.
Follow the link to the slide. Then click on the figure to play the animation. A Figure Figure
CHAPTER 4 THE DEFINITE INTEGRAL.
1 Fundamental Theorem of Calculus Section The Fundamental Theorem of Calculus If a function f is continuous on the closed interval [a, b] and F.
Riemann Sums and the Definite Integral Lesson 5.3.
The Antiderivative Safa Faidi. The definition of an Antiderivative A function F is called the antiderivative of f on an interval I if F’(x) =f(x) for.
The Fundamental Theorem of Calculus Inverse Operations.
Finding area of polygonal regions can be accomplished using area formulas for rectangles and triangles. Finding area bounded by a curve is more challenging.
The Integral chapter 5 The Indefinite Integral Substitution The Definite Integral As a Sum The Definite Integral As Area The Definite Integral: The Fundamental.
Aim: What is the Fundamental Theorem of Calculus?
Chapter 5 .3 Riemann Sums and Definite Integrals
Georg Friedrich Bernhard Riemann
The Fundamental Theorem of Calculus Lesson Definite Integral Recall that the definite integral was defined as But … finding the limit is not often.
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.
Section 5.4a FUNDAMENTAL THEOREM OF CALCULUS. Deriving the Theorem Let Apply the definition of the derivative: Rule for Integrals!
4.4c 2nd Fundamental Theorem of Calculus. Second Fundamental Theorem: 1. Derivative of an integral.
Section 4.3 – Riemann Sums and Definite Integrals
1 §12.4 The Definite Integral The student will learn about the area under a curve defining the definite integral.
4.4 The Fundamental Theorem of Calculus If a function is continuous on the closed interval [a, b], then where F is any function that F’(x) = f(x) x in.
CHAPTER 4 SECTION 4.4 THE FUNDAMENTAL THEOREM OF CALCULUS.
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