To set the stage, leverage the work we did on today’s Warm-up to pose questions that students will need to consider with other problems throughout today’s lesson. The graph provided in the warm-up problem made the limits of integration obvious to students, but they will need to compute the intersection points on their own for other problems. Additionally, students will need to consider how the calculus changes if the x-axis is above, below, or even passing through the region. To address these issues, ask students questions like:
1) How would this problem be different if both functions were shifted up 5 units?
2) What if both functions were shifted down 5 units?
3) What if both functions were shifted down 1 unit?
4) What if only the parabola were shifted up 5 units?
Through wrestling with Questions #1-3, most students will realize that the area between these functions should not change based on where the x-axis is located. The actual space between the functions does not change if both functions are translated in the same way, so the area should remain constant. Some students will argue that the region in Question #2 should be negative area because the entire region lies below the x-axis, and for Question #3 the portion of the region below the x-axis should be considered negative area and offset that same amount of positive area elsewhere above the x-axis. We will address these arguments in the Investigation section of today’s lesson, but for right now give students the opportunity to formulate arguments in support of whichever way they feel these questions should be answered (SMP #3 Construct viable arguments and critique the reasoning of others).
Question #4 raises the issue of students having to compute the new intersection points as the limits of integration. Students often say that the area of the region will get larger, which is true but is not the purpose of this question, so be sure to elicit additional student responses to note that the limits of integration will change with the new intersection points.
Prior Lesson: Approximating Definite Integrals from Tables