Estimating the area under a curve - Higher

The area under a curve can be estimated by dividing it into triangles, rectangles and trapeziums.

If we have a speed-time or velocity-time graph, the distance travelled can be estimated by finding the area.

Example

The velocity of a sledge as it slides down a hill is shown in the graph.

Find the distance travelled by the sledge over its 30 second journey.

Vertical lines every 4 seconds along the horizontal axis have been added and points joined to make triangles, rectangles or trapeziums that approximate to the curve.

A graph that shows the velocity in miles per second of a sledge over time in seconds. Broken down into seven sections with labels a-g.

The areas of the shapes are:

A \frac{4×5}{2} = 10

B \frac{4×(5+9)}{2} = 28

C \frac{4×(9+8.5)}{2} = 35

D \frac{4×(8.5+7)}{2} = 31

E \frac{4×(7+3)}{2} = 20

F \frac{4×(3+0.5)}{2} = 7

G \frac{(0.5×2)}{2} = 0.5

The total area is 10 + 28 + 35 + 31 + 20 + 7 + 0.5 = 131.5, so the sledge travelled approximately 131.5 m.

Understanding the meaning of the area

Page 1 showed how the units can be used to identify the meaning of the gradient: by dividing the vertical axis units by the horizontal axis units.

The meaning of the area under a graph can be found by multiplying the units.

For example, for the velocity-time graph above, m/s × s = \frac{metres}{seconds} × \frac{seconds}{1} = metres.

So the area represents distance in metres.

Example

A basic graph showing two axes: horizontal = litres per second, the vertical = Time in seconds

The units of the area will be \frac{litres}{seconds} × \frac{seconds}{1} = litres

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