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.
The areas of the shapes are:
A \(\frac{4脳5}{2} = 10\)
B \(\frac{4脳(5+9)}{2} = 28\)
C \(\frac{4脳(9+8.5)}{2} = 35\)
D \(\frac{4脳(8.5+7)}{2} = 31\)
E \(\frac{4脳(7+3)}{2} = 20\)
F \(\frac{4脳(3+0.5)}{2} = 7\)
G \(\frac{(0.5脳2)}{2} = 0.5\)
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, \(m/s 脳 s = \frac{metres}{seconds} 脳 \frac{seconds}{1} = metres\).
So the area represents distance in metres.
Example
The units of the area will be \(\frac{litres}{seconds} 脳 \frac{seconds}{1} = litres\)