What are the x-values when using n strips from a to b?

\(x_0 = a\), \(x_1 = a + h\), \(x_2 = a + 2h\), \ldots, \(x_n = b\), where \(h = \frac{b-a}{n}\).

Is the trapezium rule formula in the formula booklet?

Yes. You do not need to memorise it — it will be provided in the exam formula booklet.

For 4 strips from x = 0 to x = 2, what is h?

\(h = \frac{b-a}{n} = \frac{2-0}{4} = 0.5\). Each strip has width 0.5.

For 4 strips, what are the x-values?

\(x_0 = 0\), \(x_1 = 0.5\), \(x_2 = 1\), \(x_3 = 1.5\), \(x_4 = 2\). Five points for four strips.

What is the formula for the trapezium rule with n strips?

\(\int_a^b y \, dx \approx \frac{1}{2} h \{(y_0 + y_n) + 2(y_1 + y_2 + \ldots + y_{n-1})\}\) where \(h = \frac{b-a}{n}\).

If you have n strips, how many y-values do you need?

\(n + 1\) y-values. For \(n\) strips you need values at \(x_0, x_1, \ldots, x_n\) (the \(n+1\) boundaries between strips).

What is h in the trapezium rule?

\(h = \frac{b-a}{n}\) — the width of each strip. It is the distance between consecutive x-values.

When does the trapezium rule give an overestimate?

When the curve is **concave down** (bends downward). The trapeziums extend above the curve, so the total area is too large.

When does the trapezium rule give an underestimate?

When the curve is **concave up** (bends upward). The trapeziums lie below the curve, so the total area is too small.

Trapezium Rule (Numerical Integration) | Pearson Edexcel International AS Maths

Q: What is the trapezium rule?

A **numerical method** of integration that approximates \(\int_a^b y \, dx\) by dividing the region into \(n\) trapeziums and summing their areas. Used when analytical integration is difficult.

Trapezium Rule (Numerical Integration)

Summary

The trapezium rule is a numerical method for approximating the area under a curve when analytical integration is difficult or impossible. It divides the region into n strips (trapeziums), approximates each strip as a trapezium, and sums their areas. The formula uses (h = \frac{b-a}{n}) and requires (n+1) y-values. More strips generally give a more accurate estimate.

1. What is the Trapezium Rule?

The trapezium rule is a numerical method of integration used to find the approximate area under a curve.

It is used when analytical methods of integration will not work (e.g. functions with no elementary antiderivative).

It approximates the area by finding the sum of the areas of trapeziums beneath the curve.

Each strip is approximated as a trapezium with parallel sides (the y-values at the strip edges).

Diagram showing trapezium rule with strips approximating area under a curve

2. The Formula

For (n) strips (trapeziums) from (x = a) to (x = b): [\int_a^b y , dx \approx \frac{1}{2} h \big{(y_0 + y_n) + 2(y_1 + y_2 + \ldots + y_{n-1})\big}]

Where (h = \frac{b-a}{n}) is the strip width.

If you have (n) strips, you need (n + 1) y-values: (y_0, y_1, y_2, \ldots, y_n) at (x_0, x_1, \ldots, x_n).

The formula is given in the formula booklet for exams.

3. How to Apply the Trapezium Rule

STEP 1: Identify (a), (b), and (n). Calculate (h = \frac{b-a}{n}).

STEP 2: Find the x-values: (x_0 = a), (x_1 = a + h), (x_2 = a + 2h), \ldots, (x_n = b).

STEP 3: Calculate the y-values: (y_0 = f(x_0)), (y_1 = f(x_1)), \ldots, (y_n = f(x_n)).

STEP 4: Substitute into the formula: (\frac{1}{2}h\big[(y_0 + y_n) + 2(y_1 + y_2 + \ldots + y_{n-1})\big]).

4. Underestimate or Overestimate?

5. Accuracy and Percentage Error

6. Examiner Tips

Trapezium Rule (Numerical Integration)

Summary

1. What is the Trapezium Rule?

The trapezium rule is a numerical method of integration used to find the approximate area under a curve.

It is used when analytical methods of integration will not work (e.g. functions with no elementary antiderivative).

It approximates the area by finding the sum of the areas of trapeziums beneath the curve.

Each strip is approximated as a trapezium with parallel sides (the y-values at the strip edges).

Diagram showing trapezium rule with strips approximating area under a curve

2. The Formula

For (n) strips (trapeziums) from (x = a) to (x = b): [\int_a^b y , dx \approx \frac{1}{2} h \big{(y_0 + y_n) + 2(y_1 + y_2 + \ldots + y_{n-1})\big}]

Where (h = \frac{b-a}{n}) is the strip width.

If you have (n) strips, you need (n + 1) y-values: (y_0, y_1, y_2, \ldots, y_n) at (x_0, x_1, \ldots, x_n).

The formula is given in the formula booklet for exams.

3. How to Apply the Trapezium Rule

STEP 1: Identify (a), (b), and (n). Calculate (h = \frac{b-a}{n}).

STEP 2: Find the x-values: (x_0 = a), (x_1 = a + h), (x_2 = a + 2h), \ldots, (x_n = b).

STEP 3: Calculate the y-values: (y_0 = f(x_0)), (y_1 = f(x_1)), \ldots, (y_n = f(x_n)).

STEP 4: Substitute into the formula: (\frac{1}{2}h\big[(y_0 + y_n) + 2(y_1 + y_2 + \ldots + y_{n-1})\big]).

4. Underestimate or Overestimate?

Depending on the shape of the curve, the trapezium rule can give an underestimate or overestimate.

For a curve that is concave down (bends downward), the trapeziums tend to lie above the curve, so the rule overestimates.

For a curve that is concave up (bends upward), the trapeziums tend to lie below the curve, so the rule underestimates.

The more strips used, the more accurate the estimate.

5. Accuracy and Percentage Error

Calculating the percentage error can tell you how accurate your estimate is.

Percentage error = (\frac{|\text{actual} - \text{estimate}|}{|\text{actual}|} \times 100%).

Doubling the number of strips typically reduces the error significantly.

6. Examiner Tips

The formula for the trapezium rule is given in the formula booklet — you do not need to memorise it.

Ensure you have exactly (n + 1) y-values for (n) strips.

Check your values of (h) and the x-coordinates before calculating y-values.