**Unit 6 develops the ideas behind integration, the Fundamental Theorem of Calculus, and Accumulation. **(CED – 2019 p. 109 – 128 ). These topics account for about 17 – 20% of questions on the AB exam and 17 – 20% of the BC questions.

**Topics 6.1 – 6.4 Working up to the FTC**

**Topic 6.1 Exploring Accumulations of Change** Accumulation is introduced through finding the area between the graph of a function and the *x*-axis. Positive and negative rates of change, unit analysis.

**Topic 6.2 Approximating Areas with Riemann Sums** Left-, right-, midpoint Riemann sums, and Trapezoidal sums, with uniform partitions are developed. Approximating with numerical methods, including use of technology are considered. Determining if the approximation is an over- or under-approximation.

**Topic 6.3 Riemann Sums, Summation Notation and the Definite Integral. **The definition integral is defined as the limit of a Riemann sum.

**Topic 6.4 The Fundamental Theorem of Calculus (FTC) and Accumulation Functions** Functions defined by definite integrals and the FTC.

**Topic 6.5 Interpreting the Behavior of Accumulation Functions Involving Area** Graphical, numerical, analytical, and verbal representations.

**Topic 6.6 Applying Properties of Definite Integrals **Using the properties to ease evaluation, evaluating by geometry and dealing with discontinuities.

**Topic 6.7 The Fundamental Theorem of Calculus and Definite Integrals ** Antiderivatives. (Note: I suggest writing the FTC in this form because it seem more efficient then using upper case and lower case *f*.)

**Topics 6.5 – 6.14 Techniques of Integration**

**Topic 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation. **Using basic differentiation formulas to find antiderivatives. Some functions do not have closed-form antiderivatives. (Note: While textbooks often consider antidifferentiation before any work with integration, this seems like the place to introduce them. After learning the FTC students have a reason to find antiderivatives. See Integration Itinerary

**Topic 6.9 Integration Using Substitution **The *u*-substitution method. Changing the limits of integration when substituting.

**Topic 6.10 Integrating Functions Using Long Division and Completing the Square **

**Topic 6.11 Integrating Using Integration by Parts (BC ONLY)**

**Topic 6.12 Integrating Using Linear Partial Fractions (BC ONLY)**

**Topic 6.13 Evaluating Improper Integrals (BC ONLY)** Showing the work requires students to show correct limit notation.

**Topic 6.14 Selecting Techniques for Antidifferentiation **This means practice, practice, practice.

**Timing**

The suggested time for Unit 6 is 18 – 20 classes for AB and 15 – 16 for BC of 40 – 50-minute class periods, this includes time for testing etc.

**Previous posts on these topics include:**

**Introducing the Derivative**

The Old Pump and Flying to Integrationland Two introductory explorations

The Definition of the Definite Integral

The Fundamental Theorem of Calculus

Trapezoids – Ancient and Modern On Trapezoid sums

Good Question 9 – Riemann Reversed Given a Riemann sum can you find the Integral it converges to? A common and difficult AP Exam problem

**Accumulation**

Good Question 8 – or Not? Unit analysis

AP Exams Accumulation Question A summary of accumulation ideas.

Accumulation and Differential Equations

**Techniques of Integrations (AB and BC)**

Good Question 13 More than one way to skin a cat.

**Integration by Parts – a BC Topic**

Good Question 12 – Parts with a Constant?

Improper Integrals and Proper Areas

Math vs the Real World Why does not converge.

This is the fifth in a series of posts discussing the ten units in the 2019 Course and Exam Description. Other posts will appear during the year.

2019 CED – Unit 1: Limits and Continuity

2019 CED – Unit 2: Differentiation: Definition and Fundamental Properties.

2019 CED – Unit 3: Differentiation: Composite , Implicit, and Inverse Functions

2019 CED – Unit 4 Contextual Applications of the Derivative Consider teaching Unit 5 before Unit 4

2019 – CED Unit 5 Analytical Applications of Differentiation Consider teaching Unit 5 before Unit 4