Sometimes you may need to use the ‘trapezium rule’ to approximate the area beneath a curve which is very hard to Integrate. Let's learn about it in A-Level Maths!
Trapezium rule
Imagine we had a curve as shown to the right, and we wanted to find the area in the region indicated
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/388213_903898.jpeg)
We could split the region into strips, all of the same height (in this case 3), and work out the area of each strip as a trapezium
We could then add them together and the area would be an approximation for the area under the curve
If we want a better approximation, we just need to use more strips…
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/121048_273714.jpeg)
Let's see what the algebra would look like for using the trapezium rule in a question…
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/796033_304896.jpeg)
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/806751_340075.jpeg)
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/343805_484358.jpeg)
The rule
As a general case, the trapezium rule looks like this:
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/75416_51522.jpeg)
and h = (b - a) / n
The height of each strip is given by the difference between the limits, divided by‘n’, the number of strips…
Example:
Using 4 strips, estimate the area under the curve:
y=√(2x+3)
Between the lines x = 0 and x = 2
You will not need to integrate at all to do this
Start by finding the height of each strip…
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/338839_185087.jpeg)
Now draw up a table and work out y values at the appropriate x positions between 0 and 2…
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/703521_640252.jpeg)
For each of these values of x, calculate the value of y by substituting it into the equation of the curve
Now sub the values you worked out into the formula – the first value for y is y0 and the last is yn
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/592444_34028.jpeg)
Another example:
Using 8 strips, estimate the area under the curve:
y=√(2x+3)
Between the lines x = 0 and x = 2
Height = (2 - 0) / 8
=0.25
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/377842_660461.png)
![broken image](http://custom-images.strikinglycdn.com/res/hrscywv4p/image/upload/c_limit,fl_lossy,h_9000,w_1200,f_auto,q_auto/4979856/563613_252503.jpeg)
Note that this will be a better estimate as the area was split into more strips!
Drafted by Eunice (Maths)