01 · Endpoints or interior?
For five strips, which ordinates receive weight 1?
Hint
There are six ordinates, indexed from 0 to 5.
Worked solution
y₀ and y₅. The four interior ordinates receive weight 2.
Understand · explore · practise
Trapezium rule
Use the trapezium rule with equal strips, tables and missing ordinates. Understand endpoint weights, choose the correct step width and keep exact and approximate integrals distinct.
Before you startDefinite integrals, function values and areas of trapezia.
01 / Replace the curve with straight chords
A table of heights can estimate an integral.
T = (h/2)[y₀ + yₙ + 2(y₁ + ⋯ + yₙ₋₁)]
Use n equal strips of width h = (b − a)/n and n + 1 ordinates.
Join neighbouring points on the curve with straight lines. The region under each line segment is a trapezium when the heights are nonnegative. Adding these contributions approximates the integral. Written as signed contributions, the same formula also applies to negative heights.
Add the trapeziaExplore
2 strips, 3 ordinates, h = 1.
Ordinates: 1, 2, 5.
T = 1/2 × [1 + 5 + 2 × 2] = 5.
Individual trapezium contributions: 1.5 + 3.5.
The blue curve is the function. Green straight segments form the trapezia; they are approximations, not the original curve. The exact integral is 14/3. Displayed contributions are rounded; the total uses full working precision.
02 / Derive the endpoint and interior weights
Interior heights belong to two adjacent strips.
Combine three strips of width h.Worked example
Strip areas: h(y₀ + y₁)/2, h(y₁ + y₂)/2, h(y₂ + y₃)/2
Each strip uses the average of its two endpoint heights.
Total = (h/2)[y₀ + 2y₁ + 2y₂ + y₃]
Interior heights occur twice; the two outer heights occur once.
Generalise to n strips
Double every interior ordinate and keep each endpoint once.
Watch: interior heights are counted twice
Pause, replay or seek freely. The notes explain the same idea and stay in view.
03 / Count gaps rather than table entries
n strips require n + 1 sample points.
Use four strips between x = 1 and x = 3.Worked example
h = (3 − 1)/4 = 1/2
Divide the interval width by the strip count.
x-values: 1, 1.5, 2, 2.5, 3
These five points make four gaps.
Evaluate the function at all five points
Do not omit either endpoint.
02 · Six ordinates
A table has x = 0, 0.4, 0.8, 1.2, 1.6, 2. How many strips and what width?
Hint
Count the intervals between entries.
Worked solution
Five strips, h = 0.4.
03 · Find the multiplier
With six strips on [−1,2], what is h/2?
Hint
First find h = 3/6.
Worked solution
h/2 = 1/4.
04 / Build a complete arithmetic ledger
Keep the table, weights and outer factor separate.
Estimate ∫₀² (1 + x²) dx using four strips.Worked example
h = 1/2; x = 0, 1/2, 1, 3/2, 2
There are five ordinates.
y = 1, 5/4, 2, 13/4, 5
Substitute into 1 + x².
T = (1/4)[1 + 5 + 2(5/4 + 2 + 13/4)]
Only the three interior heights are doubled.
T = 19/4 = 4.75
The exact integral is 14/3, so this approximation is slightly larger.
04 · Two strips
Estimate the same integral using two strips.
Hint
Use heights 1, 2, 5 and h = 1.
Worked solution
T = (1/2)[1 + 5 + 2(2)] = 5.
05 · One strip
What estimate uses one strip on [0,2]?
Hint
There are no interior ordinates.
Worked solution
T = (2/2)(1 + 5) = 6.
05 / Use supplied data without inventing a formula
The rule only needs the measured ordinates.
At x = 0, 2, 4, 6, measured heights are 3, 5, 4, 2. Estimate the integral.Worked example
h = 2
Read the spacing from the x-values.
T = (2/2)[3 + 2 + 2(5 + 4)]
Endpoints are 3 and 2.
T = 23
The curve between measurements is unknown; this is an estimate.
06 · A data table
At x = 0, 1, 2, 3, heights are 2, 4, 3, 5. Find T.
Hint
The outer factor is 1/2.
Worked solution
T = (1/2)[2 + 5 + 2(4 + 3)] = 10.5.
07 · Check the units
If x is time in seconds and y is velocity in metres per second, what units does T have?
Hint
Multiply the horizontal and vertical units.
Worked solution
Metres. The signed integral estimates displacement; distance needs nonnegative speed or appropriate sign splitting.
06 / Solve for a missing ordinate
An interior unknown receives a doubled weight.
On x = 0,1,2,3, the heights are 2,k,4,1. The trapezium estimate is 10. Find k.Worked example
(1/2)[2 + 1 + 2(k + 4)] = 10
The unknown is an interior height.
11 + 2k = 20
Multiply by 2 and collect known terms.
k = 9/2
This is inferred from the stated estimate, not from the exact integral.
08 · Unknown endpoint
With h = 1 and heights k,3,5, the trapezium estimate is 8. Find k.
Hint
The unknown is an endpoint, so it is not doubled.
Worked solution
(1/2)[k + 5 + 2(3)] = 8 gives k = 5.
09 · Distinguish estimate from exact value
May you replace a stated exact integral by a trapezium expression and call the resulting unknown exact?
Hint
The numerical rule usually has approximation error.
Worked solution
No. Unless the problem explicitly supplies the trapezium estimate or exactness is established, the resulting value is an approximation.
07 / Use radians for trigonometric integrands
The step width must use the same variable as the integral.
Estimate ∫₀^(π/2) sin x dx with two strips.Worked example
h = π/4
Use radian values.
Ordinates: 0, √2/2, 1
Evaluate sine at 0, π/4 and π/2.
T = (π/8)(1 + √2)
Keep an exact expression for this numerical-rule estimate.
T ≈ 0.94806
The true integral is 1. An exact expression for T does not make T the exact integral.
10 · A calculator-mode check
What value should sin(π/6) give in the required mode?
Hint
The input is a radian measure.
Worked solution
1/2. Use radian mode.
08 / Check whether spacing is equal
Do not force one common h onto an uneven table.
Points x = 0,1,3 have heights 2,4,5. Estimate using straight segments.Worked example
First interval width 1: contribution (1/2)(2 + 4) = 3
Use its own width.
Second interval width 2: contribution (2/2)(4 + 5) = 9
The second interval is twice as wide.
Total = 12
Add individual trapezia; the compact equal-width formula does not apply.
11 · Identify an uneven table
Can the standard common-h formula be used directly for x = 0,1,2,4?
Hint
Compare consecutive differences.
Worked solution
No: widths are 1,1,2. Use separate widths or obtain equally spaced data.
12 · A straight-line check
Why does the trapezium rule integrate a straight line exactly?
Hint
The chord and the function coincide.
Worked solution
Every strip boundary matches the graph, so there is no curved gap to approximate.
09 / The rule estimates a signed integral
Negative ordinates can produce negative contributions.
Use one strip for y = x − 1 on [0,2].Worked example
Endpoint heights are −1 and 1
Their average is zero.
T = (2/2)(−1 + 1) = 0
This is the exact signed integral because the function is linear.
Geometric area is 1
Split at x = 1 and add two positive triangles of area 1/2.
13 · A negative constant
Use any equal strip count to integrate y = −3 over [0,2].
Hint
Every strip has the same negative height.
Worked solution
T = −6, equal to the signed integral. The geometric area is 6.
10 / Keep working values before final rounding
Premature rounding can hide an improvement.
Estimate ∫₀¹ eˣ dx using two strips.Worked example
h = 1/2; ordinates 1, e^(1/2), e
Store the calculator values without early rounding.
T = (1/4)[1 + e + 2√e]
Apply the endpoint and interior weights.
T ≈ 1.75393
Round the final estimate as requested.
14 · Two common factor mistakes
What is wrong with h[y₀ + yₙ + 2Σinterior]?
Hint
Compare the outer factor with the standard rule.
Worked solution
It is twice the trapezium estimate: the outer factor should be h/2.
15 · More data
For eight strips on [0,2], how many ordinates and what spacing are needed?
Hint
There is one more ordinate than strip.
Worked solution
Nine ordinates, h = 1/4.
11 / Count, tabulate, weight, approximate
State the result as an estimate unless exactness is justified.
- Check equal spacing before using the compact rule.
- Use n + 1 ordinates for n strips.
- Count endpoints once and interior heights twice.
- Multiply the weighted sum by h/2.
- Use radians for trigonometric functions.
- Distinguish signed integral, geometric area and approximation.
- Keep calculator precision until the final answer.
16 · Final table check
Use h = 1/2 with heights 1,2,4 to estimate the integral.
Hint
The interior ordinate is 2.
Worked solution
T = (1/4)[1 + 4 + 2(2)] = 9/4.
Section 1 of 11 · Replace the curve with straight chords