Using the Trapezium Rule to Estimate the Area Under a Curve
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The trapezium rule is a method used to estimate the area under a curve when the exact area cannot be found easily. It works by splitting the area into trapezia and adding their areas together.
Why the Trapezium Rule Is Used
For straight line graphs, the area under the graph can be found exactly using shapes such as rectangles and triangles.
For curved graphs, the shape is irregular, so the area must be estimated.
The trapezium rule improves accuracy by using straight line segments to approximate the curve.
How the Trapezium Rule Works
The interval along the x axis is divided into equal widths.
At each division, the y value is read from the graph.
Adjacent points are joined by straight lines, forming trapezia under the curve.
The total estimated area is found by adding the areas of all the trapezia.
The Trapezium Rule Formula
If the width of each strip is h, and the y values are \( y_0, y_1, y_2, \dots, y_n \), then the estimated area is given by:
$$
\frac{h}{2}\left(y_0 + y_n + 2(y_1 + y_2 + \dots + y_{n-1})\right)
$$
The first and last y values are added once.
All the middle y values are added twice.
Applying the Trapezium Rule
To use the trapezium rule:
Choose equal intervals along the x axis.
Read or calculate the corresponding y values.
Substitute the values into the trapezium rule formula.
Calculate the final area.
Using more strips gives a more accurate estimate
Estimating Area from a Graph
When y values are read from a graph, the result is an estimate rather than an exact value.
The trapezium rule is commonly used with:
• distance time graphs
• velocity time graphs
• curves in practical contexts
For example, using the trapezium rule on a velocity time graph gives an estimate of distance travelled.
Accuracy of the Estimate
The trapezium rule assumes the curve between points is a straight line.
If the curve bends sharply, the estimate may be less accurate.
Reducing the strip width makes the straight line approximation closer to the curve and improves accuracy.
Key Points to Remember
The trapezium rule estimates the area under a curve.
The area is found by adding the areas of trapezia.
All strips must have equal width.
Middle y values are counted twice in the formula.
Smaller strip widths give more accurate estimates.
The trapezium rule provides a practical and reliable way to estimate areas when working with curved graphs.