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.
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
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
In general terms, the Trapezium Rule is:
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.)
a. Complete the table below for the values of y = 15 − x2
Substituting the x-values into y = 15 − x2
|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 = (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)
The required area is 35.875 units2
Download an interactive spreadsheet (Microsoft Excel) showing the Trapezium Rule and Simpson's Rule.
(Windows users, right click and "Save target as..." to save the files on your computer.)