Topics 10.1, 10.2, 10.5
Understand that the area of a polar region is found by adding infinitely many sectors that compose the region.
Determine lower and upper limits of integration by solving for the boundary values of a particular region.
Find the area of regions bounded by a single polar curve.
Quick Lesson Plan
This lesson focuses on how to find the area of regions bounded by polar functions but is really just an extension of the idea of adding up multiple areas to find a whole (think back to Riemann sums!). Instead of presenting a formula for adding multiple sector areas, though, the students will remember the concept better if they discover the necessary integrand themselves. The first task will be to find a single sector’s area by considering it as a fraction of a circle’s area and using an angle that is the difference of the two given. Linking this idea to the formula for a sector’s area will be useful later in the lesson. Next, students will examine a limacon and will try to determine its area by estimation. Our goal is for students to remember that in Unit 6, rectangles were used to fill a region but that doesn’t seem appropriate here due to how the graph is generated (based on a radial distance from the origin, not a horizontal distance). Calculating the area of one sector will get students thinking about how multiple sectors could estimate the region’s area more accurately than rectangles could. Students will also notice that the region between two angles is not a perfect sector, but its area can be estimated using a sector, just like the area under a Cartesian curve is rarely a perfect rectangle. Once that idea has emerged, pause the class and help them to see that the area formula for a sector can be adapted to a general function based on θ. After you have debriefed page 1 of the activity, give students more time in their groups to work on questions 6-11 on the back page.
In the second part of the activity, the Geogebra applet will allow students to play with a visual representation of the function and to see what happens when the number of sectors increases, resulting in an improved estimate for the region’s area. Make sure students attend to the expression involving the sigma in the top right corner of the screen. In question 8d students will be writing an expression that represents the area of Sector k. In question 11, students use identical reasoning to Lesson 6.3 to establish that an infinite number of sectors of an infinitesimally small central angle will give the exact area of the polar region. Integrals to the rescue!
How can you find the area of a sector of a circle if you know the central angle of the sector?
Thinking about how polar functions are created, why can’t rectangles be used to find the area like we did with Riemann sums?
Why isn’t a sector a perfect fit for the limacon?
How could more sectors lead to better area estimates?
Why is sigma notation being used here?
What happens when n goes to infinity? Where have we seen this before?
The big idea here is that the integral that will produce a polar region’s area is simply the sum of multiple sectors. Students should see that the radius is based on the polar function given and the θ becomes dθ because the sectors are so small. Relating this integrand to those used for rectangular functions in Unit 6 will make connections for the students so they can realize that the same procedure is at play here, just with a different shape. The Check Your Understanding questions will allow them to gain skill in both finding these areas and also in determining which θ values should be used as limits of integration.
Students tend to struggle with knowing which θ values to use when sectors overlap such as in the limacon in CYU question 2. Try using the Geogebra applet to allow students to see the overlap and from that determine how to find the correct area. Similarly, polar graphs such as the rose can be tricky when the graph begins on the x-axis for θ = 0 but doesn’t complete a full petal. Determining when r = 0 will help them to realize that an integral that finds half of the petal’s area can simply be doubled.
Many polar functions tend to have many parentheses in them, often due to the presence of trig functions. When writing an integrand, it is important for the students to be sure to include all necessary parentheses so the integrand they present is the integrand they mean. A good idea is to not try to write complicated functions in an integrand at all, but rather to use (r(θ))^2 instead, minimizing the potential for miscommunication.
A second idea to stress with students is that “r” does indeed mean radius, but we can also think of it as “the distance from the origin”. By using these ideas interchangeably, students will be comfortable with any AP question they may encounter!