Mon Prof Augmenté
Area under a curve
Age 17 · Maths
A positive function draws a curve above a horizontal axis. The area between the curve and the axis can be approximated by many thin vertical rectangles.
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In the picture
- The smooth curve: the graph of a positive function above the baseline.
- A thin vertical rectangle: its height matches the curve at that position.
- A segment of the baseline: the small width of one rectangle.
- The shaded region: the area under the curve being approximated by the rectangles.
Examples
- Estimating the area of an irregular garden by covering it with narrow strips.
- Adding up small time slices to estimate total distance from a speed graph.
Key takeaways
- Thin rectangles approximate the area under a curve.
- More, narrower rectangles usually give a closer approximation.
Quiz: Area under a curve (Age 17)
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What does the approximation with thin vertical rectangles estimate?
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In the approximation, what does the height of each thin vertical rectangle match?
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According to the card, what usually happens when you use more, narrower rectangles?
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