Interpret and represent an infinite Riemann sum as a definite integral
I can use limits to explain how a Riemann Sum approximation can be improved to find the exact area under a curve
I can use definite integral notation to represent the exact area under a curve
Quick Lesson Plan
RRAM techniques using four subintervals are used to approximate the area beneath the cosecant curve. This familiar area expression is then adapted to accommodate an unknown number of subintervals, n, which are counted using the index i, j, or k. Finally, a limit is introduced to signify an infinite number of rectangles. This limit is then rewritten as an equivalent definite integral.
We chose to delay presenting this part of Topic 6.3 until the end of our Unit 6 study because a command of integral notation and an understanding of integral operations is necessary to fully comprehend the role of each term in a summation. Since most students probably haven’t used an index recently, a review of this concept might be necessary! This was a challenging lesson for many students. The summation expressions are complex, so using a consistent approach to dissecting a summation expression will help students tremendously. Identifying the value of (b – a) is an easy entry point for students who can then determine the lower limit of integration by examining the integrand.
Student work improved tremendously when this strategy was reviewed on subsequent days!
Students have not been required on past AP Tests to produce a summation expression from a given definite integral. Typically, a multiple-choice question presents a summation and students must identify the correct integral. Commit a short review session to this lesson before the Test in May: most students will remember the strategy used today.
Because a summation can produce an infinite number of definite integrals, expose students to a few equivalent integrals obtained by a translation of the integrand and the related shift of the limits of integration. More than one correct answer is possible!