Estimating the area under a curve

Equations and graphs Estimating the area under a curve

Below is a speed-time graph of a slow moving vehicle moving for 12 hours.

It is more difficult to find out what the area under the curve is. We can estimate it by drawing shapes that approximate the curve like this.

There are four shapes that I have used to approximate the area under the curve. A triangle on the left and then two trapeziums (the first one looks like a rectangle) and a small triangle on the right.

Let’s work out the area of these shapes.

The lower dotted line meets the y-axis at 7 and the upper dotted line meets the y-axis at 13. This leaves us with the following shapes.

The area of the first triangle is $\frac{1}{2}\times 5 \times 13 = 32.5$

The area of the first trapezium (the rectangle) is $13 \times 4 = 52$

The area of the second trapezium is $\frac{1}{2} \times (13 + 7) \times 2 = 20$

The area of the second triangle is $\frac{1}{2}\times 1 \times 7 = 3.5$

The total area of the shapes is $32.5 + 52 + 20 + 3.5 = 108$

The units are given on the axes of the graph, speed in km/h and time in hours. The distance is measured in $\frac{km}{h} \times h = km$

The estimate of the area under the curve is 108km.

Notice that this is a slight under-estimate of the area under the curve because the shapes are smaller than the area under the curve.

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