I haven’t missed Unit 7! 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 7 will post next Tuesday.

**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

Most Triangles Are Obtuse!

Half-full or Half-empty

Accumulation: Need an Amount?

AP Accumulation Questions

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

Adapting AB 1 / BC 1

**Area**

Area Between Curves

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

Adapting 2021 AB 3 / BC 3

**Volume**

Volumes of Solids with Regular Cross-sections

Volumes of Revolution

Why You Never Need Cylindrical Shells

Visualizing Solid Figures 1

Visualizing Solid Figures 2

Visualizing Solid Figures 3

Visualizing Solid Figures 4

Visualizing Solid Figures 5

Painting a Point

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

Adapting 2021 AB 3 / BC 3

**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

Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.

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 after Unit 8

2019 – CED Unit 8 Applications of Integration Consider teaching after Unit 6, before Unit 7

2019 – CED Unit 9 Parametric Equations, Polar Coordinates, and Vector-Values Functions

2019 CED Unit 10 Infinite Sequences and Series

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