A New Approach to Teaching Integrals
A typical AP Calculus course spends the first semester covering derivatives and the second semester learning about integrals. Connecting these two concepts is one of the most important skills of the course.
If you look in most textbooks, you’ll find that the integration chapter starts with information about finding antiderivatives so that students will be able to evaluate integrals. Over the course of the chapter, most textbooks will get into what the integral represents in terms of accumulation and area under the curve, as well as how to approximate definite integrals using Riemann and trapezoidal approximations. The Fundamental Theorem of Calculus is introduced early and generally from an algebraic perspective. Lots of skill-based exercises are given for practice.
When the AP Calculus CED (Course Exam Description) was released in the summer of 2019, perhaps the biggest shift occurred in Unit 6, the integration unit. Instead of starting with an analytical approach that has students evaluating, there is an emphasis on integrals representing accumulation and area under the curve (what we might consider applications of integrals). Many days are spent writing and interpreting integral expressions in context, long before students know how to evaluate these expressions. In fact, students don’t formally find antiderivatives or evaluate definite integrals analytically until more than halfway through the unit (Lesson 6.8)! Even when the Fundamental Theorem of Calculus is introduced in Lesson 6.7, students do not yet know to evaluate the antiderivative at the upper and lower limits of integration.
This shift aligns well with our own teaching philosophy: students should experience the new idea first by being introduced to it in an accessible, intuitive way, and then gradually build on this conceptual understanding with more formal notation and rigorous evaluation techniques.
Unit 6 takes us about a month to complete: the longest unit of the course (followed closely by Unit 8). In aligning our lessons with the CED we dug deeply into how the lessons were sequenced and how the ideas built on each other. We then created a complete set of EFFL lessons, review activities, and assessment suggestions for every day of the unit.
Big Ideas of Unit 6
Here’s a brief overview of the key understanding of each lesson. Understanding the progression of ideas in this unit is critical for your and your students' success.
Lesson 6.1 Area under a rate of change curve represents the accumulated quantity.
Lesson 6.2 Area under a curve can be estimated using rectangles and trapezoids whose varying heights are determined by the curve.
Lesson 6.3 An integral represents the exact area under a curve, which requires summing up infinitely many rectangles of infinitesimal width.
Note: while students should be able to use summation notation to express a sum of areas and use integral notation to represent the infinite sum, we choose to hold off on rigorously defining the rectangle height on the i-th interval. We save some of this formality for the final lesson of the unit “Returning to Riemann.”
Lesson 6.4 Accumulation functions give the area under the curve (the accumulated quantity) up to any point, x, the upper limit of integration. The derivative of an accumulation function represents how fast the quantity is being accumulated, which is the rate of change curve!
Lesson 6.5 Accumulation functions, like other functions, can be analyzed for extrema, intervals of increasing/decreasing, concavity, points of inflection, etc. by finding their first and second derivatives using the Fundamental Theorem of Calculus.
Lesson 6.6 Accumulations (represented by integrals) can be added, subtracted, multiplied by a constant, and even done in reverse (un-accumulating??)
Lesson 6.7 A definite integral of a rate of change function represents the net change in the original quantity over that time interval. If the rate of change function is a velocity function, the definite integral represents displacement, or a change in position. Evaluating a definite integral thus requires finding the difference between two outputs on the original function.
Lesson 6.8 If given a rate of change function, the original function is called an antiderivative. A function has infinitely many antiderivatives since all functions that differ by some vertical shift still have the same derivative.
Lessons 6.9, 6.10, and 6.14 Finding antiderivatives is not always straightforward and includes some advanced techniques such as u-substitution or long division.
Tips for Teaching Unit 6
Make sure students can interpret integral expressions in context and graphically before learning analytical strategies for evaluating them.
Don’t lose too much time on Riemann sum notation. Using indices and writing correct expressions for the x-value in left, right, and midpoint approximations is notoriously difficult for students, but far from the big idea of the unit. This is where a lot of students get derailed.
Encourage an attitude of strategic reasoning and problem solving when finding antiderivatives, especially when they require u-substitution. The big question is always “what function would have this function as its derivative?” Be comfortable with students using some trial-and-error instead of jumping to an algorithm they don’t yet understand.
Lessons 6.4 through 6.8 are absolutely fundamental. Assess often and focus on connecting representations and interpretation more than analytical evaluation.