AP Exam Review
Week 4  Friday (Day 20)
Focus Area(s)

Area and volume FRQs

Mixed Practice
Teaching Tips
The focus of today’s FRQs is an important application of integration: finding areas and volumes of irregular regions and solids. A quick review of topics from past AP Exams should convince you that this concept appears frequently! At this point in the year, students should be familiar not only with the expected steps for finding areas and volumes, but, also, the past AP scoring guidelines. Teachers who consistently use AP scoring guidelines develop confident students who write correct and succinct responses.
For the final 15minute FRQ review session, distribute the most appropriate problem for your student population. Each FRQ is calculatoractive (which immediately signaled to our students that they would be saving equations and coordinates in their calculator).

Use 2015 AB2 for finding areas using “upper  lower” as the integrand, calculating volume with squares as cross sections, or reviewing rates of change.

Use 2014 AB2 if working on volumes of revolution around a nonmajor axis, incorporating isosceles right triangles as cross sections, or writing original integral expressions to describe a given condition

Use 2008 AB1 when reviewing areas that require “upper  lower” as the integrand, using squares as cross sections, or finding volumes by nontraditional methods (the given area has a depth described by third function!)
And because the AP Exam was coming up fast, we discussed right away the solutions and the scoring guidelines. Not only did our students ask careful and insightful questions, but they also shared their favorite tips and hacks for navigating area and volume problems!
When facing the inevitable area and volume problems, coach your students to anticipate the following:

Storing the given equations into their calculator (if calculatoractive) and using the solving function to locate and store points of intersection; clearly naming these points (A and B, perhaps) for the readers helps to avoid copy errors, and skillfully using the numerical integration program on their calculator

Using given function names to avoid copy errors and save time when writing integral expressions

Writing integrals to represent the requested volume and, usually, a numeric answer (labels may be required); finding the antiderivative is not required on calculator active problems; if the problem is from the nocalculator section, expect the directive “write, but do not solve”

Finding volumes by slices or washers by revolving around a nonmajor axis

Finding volumes when known cross sections are stacked on a region (see 2009 AB 4b)

Writing an equation for the vertical or horizontal line that divides a given region into two equal parts

ENCOURAGE practice with the calculator (entering complex functions into Y1 and Y2, finding points of intersection, storing and recalling the points, and solving integrals without trying to find the antiderivative); using function names in the integrand; correct use of decimals before rounding a final value; reviewing area formulas

DISCOURAGE recopying a complex function by hand
Note: Finding volumes by cylindrical shells is not tested but can earn full credit when done correctly!
Teaching Tips
In this final cumulative review, we have students answer five true or false questions about key ideas of the course. We purposefully included some of the most common student errors (like forgetting the chain rule in the FTC or forgetting the correction factor with Usub!). You may wish to have students explain why a statement is false, or to correct the statement to make it true. This activity can be done individually, in small groups, or as a whole class game.
Challenge students to make their own 5 for 5 Final Review in pairs or small groups.