Area between two curves
MathematicsCalculusAges 17–18
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Calculus
Pick a pair of curves and watch the plane region between them get shaded. The simulation splits the region into thin strips to show that ∫(f − g)dx is the limit of the sum of strip areas as the number of strips grows, and it warns when the two graphs swap which one is on top within the interval, so the integral must be split by the sign of f − g.
Lesson: Geometric applications of integrals: area between curves
What it shows
The area of the region between the graphs of f and g from x = α to x = β is S = ∫|f(x) − g(x)| dx, where α and β are usually the intersection points of the two curves. The region is cut into thin vertical strips of width dx and height |f − g|; as the number of strips grows, their total area approaches the exact value of the integral, which is the meaning of a definite integral as a limit of sums. Where the curves swap which one is on top, f − g changes sign, so the integral has to be split at the intersections or taken in absolute value.
How to use
Choose the Pair of curves and, for the two parabola pairs, change Parameter k to move the intersections α and β. Start with a small Number of strips and increase it, comparing the strip total with the exact integral and watching the error shrink. Then switch to y = sin x and y = cos x and discuss why f − g must be taken in absolute value when the two graphs change places.
Parameters you can change
- Pair of curves y = x² and y = k, y = sin x and y = cos x, y = x² and y = k − x²
- Parameter k 0.5–6
- Number of strips 4–120
Questions to explore
- What is the exact area between y = x² and y = 4, and how close do 24 strips come?
- Why must the absolute value of f − g be used when finding the area between two curves?
- When k is doubled for y = x² and y = k, does the area double, or grow faster?
