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
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
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
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
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 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 7 Differential Equations Consider teaching Unit 8 before Unit 7