This unit seems to fit more logically after the opening unit on integration (Unit 6). The Course and Exam Description (CED) places Unit 7 Differential Equations before Unit 8 probably because the previous unit ended with techniques of antidifferentiation. My guess is that many teachers will teach Unit 8: Applications of Integration immediately after Unit 6 and before Unit 7: Differential Equations. The order is up to you.

**Unit 8 includes some standard problems solvable by integration **(CED – 2019 p. 143 – 161). These topics account for about 10 – 15% of questions on the AB exam and 6 – 9% of the BC questions.

**Topics 8.1 – 8.3 Average Value and Accumulation**

**Topic 8.1 Finding the Average Value of a Function on an Interval **Be sure to distinguish between* average value* of a function on an interval, *average rate of change* on an interval and the *mean value*

**Topic 8.2 Connecting Position, Velocity, and Acceleration of Functions using Integrals **Distinguish between *displacement *(= integral of velocity) and *total distance traveled *(= integral of speed)

**Topic 8. 3 Using Accumulation Functions and Definite Integrals in Applied Contexts **The integral of a rate of change equals the net amount of change. A really big idea and one that is tested on all the exams. So, if you are asked for an amount, look around for a rate to integrate.

**Topics 8.4 – 8.6 Area**

**Topic 8.4 Finding the Area Between Curves Expressed as Functions of x**

**Topic 8.5 Finding the Area Between Curves Expressed as Functions of y**

**Topic 8.6 Finding the Area Between Curves That Intersect at More Than Two Points **Use two or more integrals or integrate the absolute value of the difference of the two functions. The latter is especially useful when do the computation of a graphing calculator.

**Topics 8.7 – 8.12 Volume**

**Topic 8.7 Volumes with Cross Sections: Squares and Rectangles**

**Topic 8.8 Volumes with Cross Sections: Triangles and Semicircles**

**Topic 8.9 Volume with Disk Method: Revolving around the x– or y-Axis **Volumes of revolution are volumes with circular cross sections, so this continues the previous two topics.

**Topic 8.10 Volume with Disk Method: Revolving Around Other Axes**

**Topic 8.11 Volume with Washer Method: Revolving Around the x– or y-Axis **See Subtract the Hole from the Whole for an easier way to remember how to do these problems.

**Topic 8.12 Volume with Washer Method: Revolving Around Other Axes. **See Subtract the Hole from the Whole for an easier way to remember how to do these problems.

**Topic 8.13 Arc Length BC Only**

**Topic 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled BC ONLY**

**Timing**

The suggested time for Unit 8 is 19 – 20 classes for AB and 13 – 14 for BC of 40 – 50-minute class periods, this includes time for testing etc.

**Previous posts on these topics for both AB and BC include:**

**Average Value and Accumulation**

Average Value of a Function and Average Value of a Function

Good Question 7 – 2009 AB 3 Accumulation, explain the meaning of an integral in context, unit analysis

Good Question 8 – or Not Unit analysis

Graphing with Accumulation 1 Seeing increasing and decreasing through integration

Graphing with Accumulation 2 Seeing concavity through integration

**Area**

Under is a Long Way Down Avoiding “negative area.”

Improper Integrals and Proper Areas BC Topic

Math vs. the “Real World” Improper integrals BC Topic

**Volume**

Volumes of Solids with Regular Cross-sections

Why You Never Need Cylindrical Shells

Subtract the Hole from the Whole and Does Simplifying Make Things Simpler?

**Other Applications of Integrals**

Density Functions have been tested in the past, but are not specifically listed on the CED then or now.

Who’d a Thunk It? Some integration problems suitable for graphing calculator solution

This is the seventh 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

2019 – CED Unit 6 Integration and Accumulation of Change

2019 – CED Unit 7 Differential Equations Consider teaching Unit 8 before Unit 7