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Connecting Position, Velocity, and Acceleration using Integrals (Topic 8.2)

Unit 8 - Day 2

​Learning Objectives​
  • Determine values for positions and rates of change using definite integrals in problems involving rectilinear motion

​Success Criteria
  • I can distinguish between an object’s displacement and total distance

  • I can use integrals to connect position, velocity and acceleration

  • I can calculate position, velocity, and acceleration, displacement, and total distance using initial values

Quick Lesson Plan
Activity: Whitney’s Bike Ride



Lesson Handout

Answer Key


The familiar concepts of position, velocity, and acceleration are revisited in Topic 8.2 and a few important vocabulary terms are introduced: net distance, displacement, and total distance. To complete today’s activity, though, these concepts are connected by integration, not differentiation. 

Teaching Tips

Students who have experience in chemistry or physics have an advantage today: they are likely very comfortable with the concepts and terms surrounding displacement and motion. The informal dialogue of these students may foster a better understanding of today’s lesson for their peers. 

Communicate to students that the term total distance (and total area in Topics 8.4 - 8.6) indicates to students that all regions are considered positive: think tiling, carpeting, or painting.

Activity question 6 leads students very nicely back to their first conversations about average velocity (Unit 4), this time incorporating integrals and the Average Value Theorem, not derivatives and slopes.

The use of correct units and precise notation continues to be a focus in our classrooms: notice our labeling for Activity question 1. A label of “miles/minute squared” sometimes does not communicate to students the same information as the label “miles/minute per minute.” 

Exam Insights

The AP Exam is unlikely to ask about Whitney’s thought process (Activity question 2)! Point out that the more likely question will ask when Whitney turned around. Be sure to require a calculus explanation that includes a change in sign for her velocity.  

Student Misconceptions

Few errors in computation were observed in this lesson, but students need to be attentive to the request for “total” distance and the correct placement of their absolute value symbol. If time permits, compute an integral with the absolute value symbol placed correctly around the integrand vs the absolute value of the entire integral expression. Ask students to interpret each result. 

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