The Trapezium Rule

Areas bounded or enclosed by the graph of a function, the x-axis and vertical lines can be found by integration. They can also be found by other more approximate methods.

We study two of these methods, the Trapezium Rule and Simpson's Rule.

The red shaded area below could be approximated using the trapezium PQRS. Obviously, this would give an area that was slightly too large(extra area is shaded blue). If the curve was concave down (upside down), the approximation would be too small.

Y12_The_Trapezium_Rule_01.gif

Area of trapezium PQRS = 0.5 x height x sum of parallel sides = 0.5(b − a)(y0 + y1)

Approximating Area using Trapeziums

If the area to be found was divided into more trapeziums, the approximation would become more accurate.

Find an approximate value for the shaded area under the curve y = x2and between x = 1 and x = 4, by using three trapeziums.

Area A = 0.5 x (2 − 1) x ( 1 + 4) =2.5

Area B = 0.5 x (3 − 2) x (4 + 9) =6.5

Area C = 0.5 x (4 − 3) x (9 + 16) =12.5

Total area of trapeziums

= 2.5 + 6.5 + 12.5

= 21.5

 Y12_The_Trapezium_Rule_02.gif

This method could be speeded up by finding:

Total area of trapeziums = 0.5 x 1 x ( 1 + 2 x 4 + 2 x 9 + 16) = 21.5

Trapezium Rule

In general terms, the Trapezium Rule is:

Y12_The_Trapezium_Rule_03.gif

 

Y12_The_Trapezium_Rule_04.gif

In words Area = 0.5 x width of each trapezium x [first ordinate + last ordinate + 2(sum of all other ordinates)]

(An ordinate is a y-value. This formula is provided for the Bursary examination.)

Example
Solution

a. Complete the table below for the values of y = 15 − x2

x
0
0.5
1
1.5
2
2.5
3
y
              

Substituting the x-values into y = 15 − x2

x
0
0.5
1
1.5
2
2.5
3
y
15
14.75
14
12.75
11
8.75
6

 
b. Use the Trapezium Rule and the table of values to find the approximate area between the curve y = 15 − x2, the x-axis,the y-axis and x = 3.

The required area = Y12_The_Trapezium_Rule_05.gif (approximately)

The width of each trapezium is 0.5

Area = 0.5 x 0.5 x (15 + 6 + 2(14.75+14+12.75+11+8.75)

         = 35.875

The required area is 35.875 units2

 

Download an interactive spreadsheet (Microsoft Excel) showing the Trapezium Rule and Simpson's Rule.

button_download.gif

(Windows users, right click and "Save target as..." to save the files on your computer.)