ScientistGrades 11–12⏱ 20 min

Integrals as area

Fill the area under a curve with thinner and thinner rectangles and watch the sum close in on the integral.

1Wonder

A car’s speedometer shows its speed at every moment, but not how far it has gone. If the speed keeps changing, how can you work out the distance? The answer is an area, and the tool for finding it is the integral.

2Explore

Area under a curve

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Turn your phone sideways, or open this lesson on a tablet or computer. The explanation and quiz below work here.

Function

From a 0 To b 3 Rectangles n 6

Rectangle height from

Show the exact area

Try this

Start with 2 rectangles, then slide n up to 50. What happens to the gap between the rectangle total and the exact area?
Switch between Left, Middle and Right. For y = x², which one gives too much? Which gives too little?
Try 2x + 1. Why is the middle rule exact here, even with one rectangle?
With sin x from 0 to π, what exact area do you find?

3Explain

Step 1 of 5

Distance is an area

At a steady 3 m/s for 4 s you travel 3 × 4 = 12 m. That is the area of the rectangle under the speed–time graph. The same is true when the speed changes: distance is the area under the graph.

The definite integral

\int_a^b f(x)\,dx = \lim_{n\to\infty} \sum_{i=1}^{n} f(x_i)\,\Delta x = F(b) - F(a)

The integral is the area under f from a to b: the limit of rectangle sums, or F(b) − F(a) for any F with F′ = f.

4Practice

Question 1 of 5

A bike rides at a steady 5 m/s for 6 s. How far does it go?

5Recap

The area under a speed–time graph is the distance.
Estimate an area with rectangles; thinner rectangles give a better estimate.
The integral ∫ₐᵇ f(x) dx is the exact area, the limit of those sums.
If F′ = f, then ∫ₐᵇ f(x) dx = F(b) − F(a).