Overview

In this lesson students are first introduced to the integral and its notation. After exploring Riemann sum approximations with more and more rectangles, students explore the idea of infinitely many rectangles giving an exact area under a curve.

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Teaching Tips

Summation notation is notoriously hard for students so at this point in the unit, we have chosen to hold off on the formal notation that uses the index of summation to arrive at the left, right, or midpoint of each interval. Instead, we are using informal summation notation by referring to the rectangles from 1 to n, the height of the ith rectangle (f(xi)) and the width of each rectangle, which we are for now assuming is uniform and we are calling ∆x. Students will not see this notation during the activity portion of the lesson but the margin notes will provide this formalization.

During the activity, point out the repetitive nature of their work, adding many terms that all consist of a height and a width. Wonder aloud about a faster, more efficient way of writing this.

We suggest having students complete the first page and then debriefing the first page before turning to page 2 of the activity. We want students to really wrestle with the question of how we can get closer and closer to the exact area under a curve. Have students share out their guesses of the true area as well as the number of rectangles they would need to feel confident in their answer. If students jump straight to infinity, suggest that that would give a great estimate, but it is also awfully cumbersome.

When completing the Important Ideas, be sure to point out various parts of the integral notation and connect it back to summation notation. Once students have seen the integral for a while they tend to forget the expression under the integral symbol still represents a height and a width!

Emphasize that only when we take the limit as n (the number of rectangles) goes to infinity do we get an exact area that can be represented by the integral. This lesson mirrors students’ learning of the derivative, where we developed the idea that only when we take the limit as h goes to 0 does the average rate of change become the instantaneous rate of change that can be represented with a derivative.

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Exam Insights

Although the AP Test will not ask students to generate the summation notation in a free response question, they may be asked to recognize the structure in a Multiple Choice Question. Students should know the difference between when the sum represents an approximation (any finite number of rectangles) and when the sum represents an exact area (the limit as the number of rectangles goes to infinity).

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Student Misconceptions

New notation can be tricky for students. It is important to spend time going over all the key components of integral notation. Remind students that the limits of integration are x-values and that the integrand represents the height of each rectangle and the differential (dx) represents the width. Explain to students why ∆x becomes dx, and why it it becomes irrelevant whether a left, right, or midpoint sum is being used when the number of rectangles is infinite. We recommend assigning homework problems where students have to use an integral to represent a situation, without actually solving it by hand.