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# Area Under a Curve - Day 2 of 2

Lesson 14 of 18

## Objective: SWBAT use a sequence to estimate the area under a curve.

*40 minutes*

#### Summarize

*10 min*

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:**

- Increasing the number of rectangles will get us a more accurate estimate of the area of the shaded region.
- The base of each rectangle is the same; the only thing that will change is the height.
- 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. - We can think of the sum of the rectangles as a finite series with
*n*terms. - 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.

#### Resources

*expand content*

#### Extend

*30 min*

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.

- There was some
**confusion**about finding an estimate for the area and using a limit to find the exact area. - 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. - 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**. - 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!

#### Resources

*expand content*

- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: The Skyscraper Problem
- LESSON 2: The Fractal Tree
- LESSON 3: Describing Sequences and Series
- LESSON 4: Arithmetic Sequences
- LESSON 5: Geometric Sequences
- LESSON 6: The Fractal Tree Revisited
- LESSON 7: Investments, Loans, and Mortgages - Day 1 of 2
- LESSON 8: Investments, Loans, and Mortgages - Day 2 of 2
- LESSON 9: Mathematical Induction
- LESSON 10: Formative Assessment Review: Sequences and Series
- LESSON 11: Formative Assessment: Sequences and Series
- LESSON 12: The Limit of a Sequence
- LESSON 13: Area Under a Curve - Day 1 of 2
- LESSON 14: Area Under a Curve - Day 2 of 2
- LESSON 15: Binomial Expansion
- LESSON 16: Unit Review: Sequences and Series
- LESSON 17: Unit Review Game: Pictionary
- LESSON 18: Unit Assessment: Sequences and Series