Slicing the Space Between: Area Between Curves
Shared by A STEM Educator
Students will be able to set up and evaluate definite integrals to find the area between two curves by identifying intersection points, determining which function is on top, and computing the integral of the difference function.
Lesson Overview
Students progress from reading area under a single parabola from a graph to computing the area between two parabolas entirely from equations, changing only one variable of complexity at a time. A scavenger hunt element is woven through the task sequence so each group's answer unlocks the next card posted around the room.
Materials
- Vertical whiteboards or large chart paper
- Dry-erase markers (multiple colors per group)
- Random grouping cards
- 10 numbered scavenger-hunt task cards posted around the room (face-down until unlocked)
- QR-code answer checker stickers on the back of each card
- Graphing calculators or Desmos on devices
- Printed warm-up slips
Scaffolded Task Progression
A skate park designer sketches the cross-section of a small ramp. The shaded region below is bounded by the parabola y = 4 − x² and the x-axis.
[ Interactive graph / diagram — available in the full lesson ]
The region is shaded for you on the graph. Using the graph, state the limits of integration and write (but do not yet evaluate) the integral that gives the shaded area.
🔒 UNLOCK CODE for Card 2: The left limit of integration (as a positive integer, ignoring the negative sign) concatenated with the right limit. Example: if limits are −3 to 3, code is 33.
Same ramp cross-section as Card 1. Now EVALUATE the integral you set up:
Area = ∫ from −2 to 2 of (4 − x²) dx
[ Interactive graph / diagram — available in the full lesson ]
Give your answer as a fraction. That fraction's numerator is your unlock code for Card 3.
A skate park designer models a ramp cross-section with the curve y = -(x - 1)^2 + 9 and the ground y = 0.
[ Interactive graph / diagram — available in the full lesson ]
Find the x-intercepts algebraically, write the definite integral for the area between the curve and the x-axis, and evaluate it exactly. Give your answer as a fraction. The numerator of that fraction is your unlock code for Card 4.
Unlock the full lesson
See all 9 scaffolded tasks, the full answer key, warm-up, exit slip, and a print-ready version — free to start.
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