# Trapezium Rule

## Practise using the trapezium rule to find an approximate value for the area under a curve.

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

 Use the trapezium rule with six intervals to find an approximation of the area above the x axis and under the graph of $$y = 4x - x^2$$ between $$x=1$$ and $$x = 4$$.
 Find an approximation of the area between the x axis and the graph of $$y = \cos(x) + 2$$ between 30° and 50° using the trapezium rule with four trapeziums.
 Use the trapezium rule with five ordinates to find an approximation of$$\int^{5}_1 (x-3)^2 \; \text{dx}$$
 Use the trapezium rule with four intervals to find an approximation of$$\int^{2}_{-2} \frac{1}{x^2+1} \; \text{dx}$$
 Use the trapezium rule with five ordinates to find an approximation of the area above the x axis and under the graph of $$y = \sqrt{1+x^2}$$ for $$1 \le x \le 3$$.

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

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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

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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.

## 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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