CALCULUS · DEFINITE INTEGRAL · INVESTIGATION 20
Integration – Area Under Curve Lab
Approximate the area under y=x² over an interval using rectangles and compare with the exact definite integral.
Interactive mathematical workspace
Live simulationThe model updates as you adjust the apparatus controls. Values are simulated mathematical outputs, not physical measurements.
Primary result—
Secondary result—
Relationship—
Check—
Adjust a control to begin the investigation.
Learning objective
To investigate Integration – Area Under Curve Lab using a dynamic mathematical model and justify a conclusion from recorded values.
Mathematical principle
∫ₐᵇ x² dx = (b³−a³)/3
Procedure
1
Enter student details and note the initial settings.
2
Change one independent variable while keeping the others fixed.
3
Observe the visual model and note the calculated values.
4
Repeat with at least three different settings and record the readings.
5
Use the pattern in your observations to write a mathematical conclusion.
Variables
Independent: the control being changed. Dependent: calculated result. Controlled: other settings held constant.
Guided investigation: Increase the rectangle count and compare the left Riemann sum with the exact area.
Common errors to avoid
- Changing several variables simultaneously.
- Recording rounded values before completing calculations.
- Confusing linear, square and cubic units.
- Writing a conclusion that is not supported by observations.
Live mathematical analysis
∫ₐᵇ x² dx = (b³−a³)/3
The working will appear here.
Observation table
| Trial | Settings | Result 1 | Result 2 |
|---|---|---|---|
| Run and record trials to populate this table. | |||
Compare the readings and identify the pattern.
Digital practical notebook
Recorded trials
| # | Settings | Result 1 | Result 2 | Time |
|---|
Knowledge check
Select the best answer. Feedback is provided after checking.
Assessment score0 / 5
Investigation challenge
Estimate area under y=x² from 0 to 3.