Trapezium Rule

If we were to find the area beneath a line, the first choice would be to integrate the equation of the line with respect to x.

However, sometime the equation of the line is too complex to integrate or the question may ask you to work out the area using the trapezium rule.

The trapizium rule is a less accurate way of working out the area beneath a graph and always gives an exaggerated result but weve gotta learn it.

There is only one formula needed when answering a trapezium rule question:

h/2 (a + b)          The area of a Trapezium

Here is an exam question from MEI Core 2 from January 2006.

Use the trapezium rule with six strips to estimate the area of the region bounded by the curve, the lines x=1 and x=4 and the axis 

x 1   1.5    2   2.5   3  3.5  4

y 8.2 6.4 5.5 5.0 4.7 4.4 4.2

state with a reason whether the trapezium rule gives an overestimate or an underestimate of the area of this region.                              

                                                                                                                      [5]

(In this question h = 0.5 for all the trapeziums). 

How to Answer 

1st trapezium : 0.5/2 (8.2 + 6.4) = 3.65

2nd trapezium : 0.5/2 (6.4 + 5.5) = 2.975

3rd trapezium : 0.5/2 (5.5 + 5) = 2.625

4th trapezium : 0.5/2 (5 + 4.7) = 2.425

5th trapezium : 0.5/2 (4.7 + 4.4) = 2.275

6th trapezium : 0.5/2 (4.4 + 4.2) = 2.15

sum of all trapeziums = 16.1 cm^2

The answer is an overestimate as the lines of the trapezium always cross the line of the curve even is it's only slightly