Area Under a Curve - Day 2 of 2

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Objective

SWBAT use a sequence to estimate the area under a curve.

Big Idea

How can we find the area of highly irregular regions?

Summarize

10 minutes

I start class by having students recap the work we did yesterday when we tried to find the area under a curve. There are a few big ideas that I hope they understand, so if I don't fully get the impression that they have a grasp on these, I will ask questions to bring them out. The big ideas are:

  1. Increasing the number of rectangles will get us a more accurate estimate of the area of the shaded region.
  2. The base of each rectangle is the same; the only thing that will change is the height.
  3. To get the height of the rectangle we plug each x-value into the function. The x-values will always be equally spaced out and will increase by w/n where w is the width of the entire region and n is the number of rectangles.
  4. We can think of the sum of the rectangles as a finite series with n terms.
  5. To get the exact area of the region, we need to divide the region into an infinite number of rectangles. Since this is impossible, we can find the limit to give us the exact answer.

In this video I talk about what my true intent was for this lesson and yesterday's. This is also something I communicate with my students before they start working on the assignment. 

Extend

30 minutes

After recapping yesterday's work, I give students this worksheet for them to work on with their table groups. While we are only working on one specific type of problem (finding the area under a curve), it is a challenging task and I want them to have practice going through the steps of making an infinite number of rectangles. I give my students about 30 minutes to work on this in class with their groups.

Here are a couple things that I noticed as they were working on the assignment.

  1. There was some confusion about finding an estimate for the area and using a limit to find the exact area.
  2. There were some calculator issues. Some students were having difficulty with the technical aspects of plugging finding a sum on a calculator. Missing a parenthesis or having an extra one were very common. 
  3. For question #1a) the estimates were less than the actual area. Yesterday the estimates were greater than the actual area. I had a discussion with a few students about how this relates to whether or not the curve is increasing or decreasing
  4. Everyone was very engaged while working on this - more so than usual. All students were in the zone! Students were going to other tables to compare with other people in the class. I can't put my finger on why but my gut tells me that it was because the level of difficulty was at the perfect level. This is a challenging problem with many different concepts, but we were basically only using areas of rectangles. Whatever the reason, I was really happy to see it!