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Using Accumulation and Definite Integrals in Applied Contexts (Topic 8.3)

Unit 8 - Day 3

​Learning Objectives​
  • Interpret the meaning of a definite integral in accumulation problems

  • Determine net change using definite integrals in applied contexts

​Success Criteria
  • I can interpret the meaning of a definite integral as an accumulation of a quantity or a change in the quantity

  • I can determine net change using the FTC for definite integrals

  • I can solve rate in/rate out problems

Quick Lesson Plan
Activity: How many People are at the Met?



Lesson Handout

Answer Key


The Metropolitan Museum of Art is the focus for today’s calculus work!  By examining classic “rate in” and “rate out” expressions, students will compute the rate at which visitors are entering the museum, the number of people who have entered during a specified time interval, and the average rate at which visitors enter the museum. Students will compare and combine “rate in” and “rate out” values to find extrema and then revisit initial values and accumulation functions to find the number of people in the museum at a given time.. 

Teaching Tips

Most students will work easily through questions 1a - 1c in the opening Activity. Notice the integrand in question 1c: using M(t) is acceptable when the function M(t) has been defined.  The result in Activity question 2 may also be justified with an inequality that shows M(4) - L(4) > 0. See also CYU 1b. 

Point out to students that while Activity question 3 requires the usual justification for a maximum, the ROC function actually contains two rate expressions: M(t) and L(t). See also CYU 1c. 

The FTC appears in Activity question 4 where one initial value (25 people) combines with two accumulation functions to calculate the number of people in the museum at closing time. See also CYU 1d. 

Exam Insights

Using technology efficiently can be a challenge for even the most talented calculus student. We used today’s activity to emphasize features on the calculator that can save time during the AP Exam. Encourage students to frontload relevant equations into Y1 and Y2 at the start of an extended problem to better utilize MATH:8 (differentiation), MATH:9 (integration), and graphing utilities on their calculators. 

Student Misconceptions

Attention to domain restrictions plays an important role in Activity question 4. The function M(t) is not defined on the same domain as L(t), so two separate integrals are required for solving.
CYU 1a provides practice with a graphical representation to reinforce the connection between an accumulation function (integrals) and area beneath a ROC curve. 

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