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Calculus AB - Unit Three

The Integral and the Fundamental Theorem of Calculus

 

Integrals are used in a wide variety of practical and theoretical applications. AP Calculus students should understand the definition of a definite integral involving a Riemann sum, be able to approximate a definite integral using different methods, and be able to compute definite integrals using geometry. They should be familiar with basic techniques of integration and properties of integrals. The interpretation of a definite integral is an important skill, and students should be familiar with area, volume, and motion applications, as well as with the use of the definite integral as an accumulation function. It is critical that students grasp the relationship between integration and differentiation as expressed in the Fundamental Theorem of Calculus — a central idea in AP Calculus. Students should be able to work with and analyze functions defined by an integral.

Click on a link to jump ahead to the following sections:

  • Lessons
  • Other Resources


Lessons

  • Unit 1: Introduction to Integrals
    • 3.11 Can I Get An Inverse? - Introduction to Integral Calculus
    • 3.12 Area - What Can Area Represent?


  • Unit 2: The Definite Integral
    • 3.21 Riemann Sums - Estimating the Area Under/Over a Curve
    • 3.22 Definite Integral - Just Like Derivatives…but Backwards
    • 3.23 The Definite Integral - an introduction to the notation of integrals and the concept of area under the curve (from College Board)
      • 3.23b Interpreting Notational Expressions - FRQ from AP Central
    • 3.24 Definite Integral - TI Nspire lesson (alternative to 3.23b that uses TI Nspire calculator
      • 3.24a Definite Integral - .tns file



  • Unit 3: The Fundamental Theorem of Calculus
    • 3.31 – Invoking a Higher Power - the Fundamental Theorem of Calculus
    • 3.32 A Deeper Dive - Finding Antiderivatives and Evaluating Functions Defined by Integrals
    • 3.33 FTC2 - Part 2 of the Fundamental Theorem of Calculus
    • 3.34 Nothing Compares to U - Integration by Substitution and the Reverse Chain Rule
    • 3.35 Interpreting Definite Integrals - an introduction to applications of Integration by Substitution (College Board)
      • 3.35b Related Rates - FRQ from the College Board

  • Unit 4: Differential Equations
    • 3.41 Changing Gears - Introduction to Differential Equations
    • 3.42 Qualitative Behavior in Differential Equations - Slope Fields, Equilibrium Solutions and more…
    • 3.43 I’m Waking Up to Ash and Dust… - Separable Differential Equations
    • 3.45 In the Real World - Modeling Differential Equations
    • 3.45 It’s a Wild, Wild World - Population Growth and Logistic Equations/li>


  • Unit 5: Applications
    • 3.51 Average Value - Finding the average value of a function on an interval
    • 3.52 Particle Movement - Finding arc lengths of curves given by parametric equations
    • 3.53 Applications - Using accumulation functions and definite integrals in applied contexts
    • 3.54 Area - Using Definite Integrals to Find Area
    • 3.55 Volume - Using Definite Integrals to Find Volume of Solids

  • Unit 6: Assessment Tasks
    • FRQ - Interpreting Context - FRQ from AP Central
    • Cooperative Task - Slope Fields Card Matchg
    • Project - Group or Individual Task

  • Supplemental Tasks:
    • 3.36 Cooperative Task Sort-the-Steps for Integration by Substitution (from College Board)





Resources - Integrals

The following are links to resources for Integrals:

  • Big Idea 3 - An overview of the Integrals module from the College Board

  • Integration and accumulation of change - Entire unit on Khan Academy

  • Differential Equations - Entire unit on Khan Academy

  • Applications of Integration - Entire unit on Khan Academy



The following are other free resources:

  • GeoGebra - Free online math tools

  • Desmos - Free online graphing calculator






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