Calculate areas in the plane using the definite integral
I can calculate areas between curves expressed as functions of y
I can find intersections of curves to determine the limits of integration
I can use a sum of multiple definite integrals when curves intersect at more than two places
Quick Lesson Plan
We continue our study of the area between curves by considering the planks on a deck. Using strategies from their Topic 8.4 and 8.6 work, vertical rectangles transition to horizontal rectangles when curves are expressed as functions of y. In order for students to develop this concept independently, they are asked to compare the x- and y-intervals that bound the deck. They then determine that using horizontal rectangles to find area (as opposed to the familiar vertical rectangles) requires only a single integral!
Model a step-by-step process and precise notation when solving for x: label the new function as g(y), perhaps, and remember to discuss the positive and negative square root assignment if the original f(x) is a quadratic expression.
Our vocabulary changed from “upper minus lower” to “right minus left” with a careful definition of what constituted the “right” function. In the upcoming lessons of volumes of solids of revolution we transition to the language of “further right minus closer” to determine the radii length. You could introduce that vocabulary in this lesson to keep consistent language.
The points of intersection in Activity #7 are simple, but CYU #1 reminds students to store values in their calculator for easy recall later.
Finding the area of a region between curves is a staple of the AB exam. Questions are found on both the calculator and non-calculator sections. When this question appears on the calculator active section, students must be sure to write the definite integral expression (including limits) along with the value produced by their calculator. Writing an antiderivative expression would not be required, although some students may attempt to find it. This is not an efficient use of the test-taking time and students may actually lose points for an incorrect antiderivative! As calculators become more sophisticated, look for these questions to appear more frequently on the non-calculator section of the exam.
Remind students that regions must be closed and bounded and that the area, regardless of the quadrant or location above or below the x-axis, will be a positive quantity.
Encouraging students to draw horizontal rectangles within the shape will help them choose to use a “dy” integral expression. Determining the most efficient method for finding an area (dx vs. dy) is a skill that is developed after much practice, but will lead to increased efficiency later on the AP Exam.
Some students will write a “dy” integrand, but use an x-interval when integrating, or vice versa. Again, drawing rectangles can help clarify this task. Then ask students for the smallest and largest y-values which create the rectangles in the region of interest.
We hope that students solve CYU #2 with a dy integral, but attempting a dx solution can be a valuable exercise. Knowing how to integrate from x = -3 to x = -2 requires solving for y and using both the positive and negative root in the integrand.