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Finding Areas between Curves Expressed as Functions of x (Topics 8.4 and 8.6)

Unit 8 - Day 5

​Learning Objectives​
  • Calculate areas in the plane using the definite integral

​Success Criteria
  • I can calculate areas between curves expressed as functions of x

  • 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
Activity: How Rich are the Top 1%?



Lesson Handout

Answer Key


In this lesson students begin their study of area between curves by looking at the Lorenz function, which highlights how income is distributed among the population. Students consider how the area of the given region represents income inequality and use this to calculate the Gini Coefficient. In today’s lesson area is found using vertical rectangles when curves are expressed as functions of x. We will discuss horizontal rectangles tomorrow. 

Teaching Tips

In a bit of good fortune, Presidential candidate Bernie Sanders was scheduled to visit our city two days after we presented this lesson. With a focus on income inequity, the lesson for Topics 8.4 and 8.6 was well-timed! Today’s lesson should be presented without revealing your personal opinions about politics or world economies. However, our students were very engaged and willing to share information from their AP Econ, Dystopian Lit, and AP Gov classes!

Before presenting this lesson, we had a lengthy conversation about the interpretation of the area between the curves for economic equality (y = x) and the Lorenz function (L(x)). For calculus to remain the focus of the lesson, we agreed that the area between the graphs would represent the accumulated economic disadvantage, or the accumulated economic inequality, of a society. 

Our students discussed variations on the given Lorenz graph: what does the area A represent when L(x) is almost linear? Can L(x) be even more exaggerated? Can L(x) ever exist above the line y = x?

By developing the formula for area between curves independently in Activity question 5, the formalization of the equations in the Important Ideas section went smoothly.

Exam Insights

Finding the area of a region between two curves is a staple of the AB exam. Questions are found on both the calculator and non-calculator sections. Students will need to know trig values for many of the non-calculator questions! 

For valuable FRQ practice, serve up the following: 2019 5a (trig! no calculator!), 2015 2a (requires two regions, calculator allowed), 2014 2c (tricky!), 2013 5a (no calculator for this one!), 2012 2a (calculator allowed), 2011 3b (trig --- calculator allowed, though), 2011 3b (trig again), 2011 3a (on Form B, requires two regions), or 2010 4a. 

Student Misconceptions

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 vertical rectangles within the shape (horizontal rectangles coming soon…) will help them choose to use a “dx” integral now and a “dy” integral later. 

The points of intersection in CYU question 2 are intentionally messy. Help your students store these values in their calculator and then show them how to “recall” the values once they access MATH:9. This is a useful skill to learn and practice now so they can save precious time on the AP Exam. 

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