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Week 4 - Friday (Day 20)

All Weeks
Focus Area(s)
• Area and volume FRQs

• Mixed Practice

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Materials: 2015 AB2 or 2014 AB2 or 2008 AB1

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 15-minute FRQ review session, distribute the most appropriate problem for your student population.  Each FRQ is calculator-active (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 non-major 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 non-traditional 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 calculator-active) 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 no-calculator section, expect the directive “write, but do not solve”

• Finding volumes by slices or washers by revolving around a non-major 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

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Note: Finding volumes by cylindrical shells is not tested but can earn full credit when done correctly!

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