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      • 2-1/2-2 Solving Equations/Inequalities
      • 2-3 Linear Functions and Slope-Intercept Form
      • 2-4 Families of Functions
      • 2-5 Absolute Value Functions and Graphs
      • 2-6 Two-Variable Inequalities
      • 2-7 Solving Systems of Equations
      • 2-8 Systems of Inequalities
      • 2-9 Systems With Three Variables
      • Chapter 2 Reviews
    • Chapter 3 - Quadratic Functions and Equations >
      • 3-1 Quadratic Functions and Transformations
      • 3-2 Standard Form of a Quadratic Function
      • 3-3 Factoring Quadratic Expressions
      • 3-3b Intercept Form
      • 3-4 Quadratic Equations
      • Graphing Quadratics Handouts
      • 3-5 Completing the Square
      • 3-6 The Quadratic Formula
      • Chapter 3 Reviews
    • Chapter 4 Polynomials and Polynomial Functions >
      • 4-1 Polynomial Functions
      • 4-2 Adding, Subtracting, and Multiplying Polynomials
      • 4-3 Polynomials, Linear Factors, and Zeros
      • 4-4 Solving Polynomial Equations
      • 4-5 Dividing Polynomials
      • 4-6 Theorems About Roots of Polynomial Equations
      • Chapter 4 Handouts
      • Chapter 4 Reviews
    • Chapter 5 - Rational Expressions and Functions >
      • 5-1 Simplifying Rational Expressions
      • 5-2 Multiplying and Dividing Rational Expressions
      • 5-3 Adding and Subtracting Rational Expressions
      • 5-5 The Reciprocal Function Family
      • 5-7 Solving Rational Equations
      • Chapter 5 Handouts
      • Chapter 5 Reviews
    • Chapter 6 - Radical Expressions and Functions >
      • 6-1 Roots and Radical Expressions
      • 6-2 Multiplying and Dividing Radical Expressions
      • 6-3 Binomial Radical Expressions
      • 6-4 Rational Exponents
      • 6-5 Solving Square Root and Other Radical Equations
      • 6-6 Functions Operations
      • 6-7 Inverse Relations and Functions
      • 6-8 Graphing Radical Functions
      • Chapter 6 Handouts
      • Chapter 6 Reviews
    • Chapter 7 - Exponential and Logarithmic Functions >
      • 7-1 Exploring Exponential Models
      • 7-2 Properties of Exponential Functions
      • 7-3 Logarithmic Functions as Inverses
      • 7-4 Properties of Logarithms
      • 7-5 Exponential and Logarithmic Equations
      • 7-6 Natural Logarithms
      • Chapter 7 Handouts
      • Chapter 7 Reviews
    • Chapter 8 - Trigonometric Functions >
      • 8-0 Right Triangle Trigonometry
      • 8-1 Exploring Periodic Data
      • 8-2 Angles and the Unit Circle
      • 8-3 Radian Measure
      • Chapter 8 Handouts
      • Chapter 8 Reviews
    • Spring 2019 Final Project
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Chapter 6 - The Definite Integral

6.1 - Estimating with Finite Sums
6.2 - Definite Integrals
6.3 - Definite Integrals and Antiderivative
6.4 - The Fundamental Theorem of Calculus
6.5 - Trapezoidal Rule
Chapter 6 Reviews

The need to calculate instantaneous rate of change led the discoveries of calculus to an investigation of the slopes of tangent lines and, ultimately, to the derivative--to what is called differential calculus.  But derivatives revealed only half the story.  In addition to a calculation method (a "calculus") to describe how functions change at any given instant, they needed a method to describe how those instantaneous changes could accumulate over an interval to produce the function that describes the total change.

Early in the 14th century, a group of scholars in Oxford, England, explored how to find the accumulated distance from knowledge of the velocity.  In 1638, Galileo used their ideas to explain the motion of falling bodies and justify his claim that the earth circles the sun.  Later in that century, Isaac Newton studied these accumulation functions, and in 1666, discovered a remarkable insight that connected accumulation functions to what he already knew about derivatives.  If there was one moment when calculus was born, this was it.

Today, what we call the integral calculus or integration has two distinct interpretations.  We begin this chapter by looking at integration as accumulation.  but is also can be viewed as reversing the process of differentiation, what we call antidifferentiation.  Newton's insight, that these two are connected, is what is called the Fundamental Theorem of Calculus.  (Taken from an excerpt in the fifth edition of the Finney, Demana, Waits, Kennedy, and Bressoud textbook Calculus - Graphical, Numerical, Algebraic.)


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  • Home
  • Integrated Math 3
    • Homework Calendars
    • Chapter 2 - Linear Equations, Inequalities, and Functions >
      • 2-1/2-2 Solving Equations/Inequalities
      • 2-3 Linear Functions and Slope-Intercept Form
      • 2-4 Families of Functions
      • 2-5 Absolute Value Functions and Graphs
      • 2-6 Two-Variable Inequalities
      • 2-7 Solving Systems of Equations
      • 2-8 Systems of Inequalities
      • 2-9 Systems With Three Variables
      • Chapter 2 Reviews
    • Chapter 3 - Quadratic Functions and Equations >
      • 3-1 Quadratic Functions and Transformations
      • 3-2 Standard Form of a Quadratic Function
      • 3-3 Factoring Quadratic Expressions
      • 3-3b Intercept Form
      • 3-4 Quadratic Equations
      • Graphing Quadratics Handouts
      • 3-5 Completing the Square
      • 3-6 The Quadratic Formula
      • Chapter 3 Reviews
    • Chapter 4 Polynomials and Polynomial Functions >
      • 4-1 Polynomial Functions
      • 4-2 Adding, Subtracting, and Multiplying Polynomials
      • 4-3 Polynomials, Linear Factors, and Zeros
      • 4-4 Solving Polynomial Equations
      • 4-5 Dividing Polynomials
      • 4-6 Theorems About Roots of Polynomial Equations
      • Chapter 4 Handouts
      • Chapter 4 Reviews
    • Chapter 5 - Rational Expressions and Functions >
      • 5-1 Simplifying Rational Expressions
      • 5-2 Multiplying and Dividing Rational Expressions
      • 5-3 Adding and Subtracting Rational Expressions
      • 5-5 The Reciprocal Function Family
      • 5-7 Solving Rational Equations
      • Chapter 5 Handouts
      • Chapter 5 Reviews
    • Chapter 6 - Radical Expressions and Functions >
      • 6-1 Roots and Radical Expressions
      • 6-2 Multiplying and Dividing Radical Expressions
      • 6-3 Binomial Radical Expressions
      • 6-4 Rational Exponents
      • 6-5 Solving Square Root and Other Radical Equations
      • 6-6 Functions Operations
      • 6-7 Inverse Relations and Functions
      • 6-8 Graphing Radical Functions
      • Chapter 6 Handouts
      • Chapter 6 Reviews
    • Chapter 7 - Exponential and Logarithmic Functions >
      • 7-1 Exploring Exponential Models
      • 7-2 Properties of Exponential Functions
      • 7-3 Logarithmic Functions as Inverses
      • 7-4 Properties of Logarithms
      • 7-5 Exponential and Logarithmic Equations
      • 7-6 Natural Logarithms
      • Chapter 7 Handouts
      • Chapter 7 Reviews
    • Chapter 8 - Trigonometric Functions >
      • 8-0 Right Triangle Trigonometry
      • 8-1 Exploring Periodic Data
      • 8-2 Angles and the Unit Circle
      • 8-3 Radian Measure
      • Chapter 8 Handouts
      • Chapter 8 Reviews
    • Spring 2019 Final Project
  • About
  • Contact