Interpret the meaning of area under a rate of change function in context.
Approximate area under a curve using geometric and numerical methods.
Interpret and represent an infinite Riemann sum as a definite integral.
Quick Lesson Plan
In preparation for Quiz 6.1 – 6.3, today’s review activities focus on the skills and content of Topics 6.1 – 6.3: the concept of accumulation, approximations using Riemann sums and trapezoidal sums, and the development of the integral as a limit of approximating Riemann sums. We provided a wide variety of tasks for students, starting with an in-depth investigation of the area underneath the sine curve. While students do not yet know how to evaluate definite integrals using the Fundamental Theorem of Calculus, they can reason geometrically using the symmetry of the curve. The primary purpose of this activity is to solidify students' understanding of integral notation and its graphical interpretation, and to preview upcoming ideas about integral properties. Students apply what they know about transformations, even and odd functions, and geometry to evaluate a variety of integrals, with an emphasis on providing a convincing argument for why their answer is correct. Page 2 provides a series of challenge questions to further students' explorations.
Next, we gave students the "Cookie Craze" problem to work on. With many FRQ-like problem stems, students practice interpreting and approximating definite integrals. We also ask students to compare approximation methods, even though students can not say definitively which methods provide overestimates and which provide underestimates, since the function is not strictly increasing or decreasing. Instead, we want students reasoning about which data points are used (and not used) in each method. Additionally, students should understand why a trapezoidal approximation will always produce better approximations than a rectangular (Riemann) approximation. Part f) previews the idea of an accumulation function. Avoid providing formal instruction on this topic and just see what students will do. Many of our students were able to successfully write an integral expression where the upper limit of integration was w.
Finally, we gave students some original “Open Middle” integral expressions to encourage exploration of different integrands. Students complete integral expressions using the digits 1-9 and must consider how the degree of the polynomial function, the horizontal translation, and the limits of integration all affect the value of the integral. These problems are challenging but provide rich opportunities for reasoning, sense-making, and discussion. If you are unfamiliar with these types of problems, you can learn more here.
Encourage student-to-student conversation and group work during these review activities as students will benefit from the informal instruction of their peers! If your class finishes early with these activities, you can distribute FRQs from past AP Tests which reflect the learning objectives and skills from Topics 6.1-6.3 or allow time for students to complete previously assigned homework problems.