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Trapezium Rule

Practise using the trapezium rule to find an approximate value for the area under a curve.
Menu Level 1 Level 2 Level 3 Exam-Style Help Calculus

Answers with more than three significant figures should be rounded to three significant figures.

This exercise will take you step by step through the process of approximating the area of the curve defined by the following function:

$$y=\frac{24}{\left(x+5\right)}+3$$

The area is required between \(x=1\) and \(x=5\) and will be found by splitting this area into 4 regions that can be roughly drawn as trapeziums.

Diagram 1

Complete the following table to show the ordinates, the lengths of the vertical sides of the trapeziums.

\(x\)12345
\(y\) Correct Wrong Correct Wrong Correct Wrong Correct Wrong Correct Wrong

The four trapeziums can be numbered one to four from left to right. Calculate the areas of each of the trapeziums:

Trapeziumnumber 1number 2number 3number 4
Area Correct Wrong Correct Wrong Correct Wrong Correct Wrong

Add the areas together to find the total area of the green area under the curve.

Correct Wrong

 

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This is Trapezium Rule level 1. You can also try:
Level 2 Level 3

Instructions

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Description of Levels

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Level 1 - A structured single question with many parts

Level 2 - Five practice questions

Level 3 - Questions requiring a little more thought

Exam-Style - Have a go at an IB/A-level exam-style questions (worked solutions are available for Transum subscribers).

Calculus a collection of lesson Starters, visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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Example

The video above comes from Revision Village

Sometimes the area under a curve cannot be found by integration. In these cases a method to approximate the area under the curve called the trapezium rule can be used.

The rule divides the area under a curve into trapeziums, calculates their areas, then sums these areas to get an approsimation of the total area.

Area of a trapezium

Formula for the area of a trapezium

Trapezium Rule

If the width of each interval is \(h\) and the y values (ordinates) are denoted as \(y_0, y_1, y_2, y_3 ...\) then the formula for finding the sum of the areas of these trapeziums is

$$ \frac12 h ((y_0 + y_n) + 2(y_1 + y_2 + ... + y_{n-1})) $$

where n is the number of intervals.

Note that the number or ordinates is always one more than the number of intervals.

If the lower bound of the required area is \(p\) and the upper bound is \(q\) then

$$ h= \frac{q-p}{n} $$

The more trapeziums the area is divided into the more accurate the estimate.

When the gradient of the graph is increasing over the given interval the area given by the trapezium rule will be an overestimate of the actual area.

When the gradient of the graph is decreasing over the given interval the area given by the trapezium rule will be an underestimate of the actual area.

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