The Trapezium Rule
The trapezium rule is a method to estimate the area under a curve by dividing the region into vertical strips and summing the areas of the trapezia formed. This is useful when direct integration is too complex or when you only have discrete data points.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The method approximates a curve with a series of straight line segments, forming the top edges of the trapezia.
- The accuracy of the estimate increases as you use more strips (i.e., as the width of each trapezium, h, decreases).
- Whether the approximation is an overestimate or an underestimate depends on the concavity (curvature) of the function.
- If the curve is concave up, the straight lines lie above the curve, resulting in an OVERESTIMATE.like y = x2
- If the curve is concave down, the straight lines lie below the curve, resulting in an UNDERESTIMATE.like y = -x2
- In the ESAT, you can always assume the strips (trapezia) have equal width.
Diagram
Formulae
Area ≈ (h/2) × [y0 + yn + 2(y1 + y2 + ⋯ + yn-1)] To estimate the area under a curve y=f(x) from x=a to x=b using n strips. Here, h = (b-a)/n is the strip width, y0 is the first y-value, yn is the last, and the y1⋯yn-1 are all the intermediate y-values.
Definitions
- Concavity
- The direction in which a curve bends. A curve that opens upwards is 'concave up', while one that opens downwards is 'concave down'.
- Ordinates
- The y-coordinates of points on a curve, used as the parallel side lengths of the trapezia in the rule.
Worked example
Use the trapezium rule with 4 strips to estimate the value of the integral of y = √(x) from x = 0 to x = 4. State whether your answer is an overestimate or an underestimate.
- 1
1.
Identify the parameters:
The interval is [a, b] = [0, 4] and the number of strips is n = 4.
- 2
2.
Calculate the width of each strip:
h = (b - a) / n = (4 - 0) / 4 = 1 - 3
3.
Determine the x-coordinates for the ordinates:
x0=0, x1=1, x2=2, x3=3, x4=4 - 4
4.
Calculate the corresponding y-values (ordinates):
y0=√(0)=0, y1=√(1)=1, y2=√(2), y3=√(3), y4=√(4)=2 - 5
5.
Apply the trapezium rule formula:
Area ≈ (h/2) × [y0 + y4 + 2(y1 + y2 + y3)].
- 6
6.
Substitute the values:
Area ≈ (1/2) × [0 + 2 + 2(1 + √(2) + √(3))].
- 7
7.
Simplify the expression:
Area ≈ (1/2) × [2 + 2 + 2*√(2) + 2*√(3)] = (1/2) × [4 + 2*√(2) + 2*√(3)] = 2 + √(2) + √(3).
- 8
8.
Determine the nature of the estimate:
The graph of y = √(x) is concave down for x > 0.
Therefore, the straight tops of the trapezia lie below the curve, resulting in an underestimate.
Answer: The estimated value is 2 + √(2) + √(3). This is an underestimate.
Common mistakes
- ×Mixing up the formula: Forgetting the factor of 2 for the inner ordinates, or applying it incorrectly to the first (y0) or last (yn) ordinate.
- ×Incorrectly determining over/underestimate: The easiest way to be certain is to sketch the curve. A curve that 'spills water' (like a hill) is concave down (underestimate), while one that 'holds water' (like a valley) is concave up (overestimate).
- ×Arithmetic errors: Simple mistakes when adding fractions or manipulating surds are common under time pressure without a calculator.
No-calculator tips
- ✓Always draw a quick sketch of the graph. It instantly shows the concavity for determining over/underestimate and helps you visualise the trapezia you are calculating.
- ✓Handle surds and fractions carefully. Don't try to approximate values like √(2) unless the question explicitly asks for a numerical estimate. Leave the answer in its exact form.
- ✓Organise your calculations in a table with columns for x, y, and whether the y-value is an end or middle one. This helps prevent mistakes when substituting into the formula.