The btrapezium rule/b can be used to find the approximate area under a curve for functions which are impossible to integrate algebraically.

Consider the graph below of (y=f(x)).

mtaimg/images/topics/9/9-102-1.png/mtaimg

The area under the curve from (x=a) to (x=b) can be split into (n) strips (trapeziums).

The sum of the areas of the trapeziums is an approximation of the area under the curve, i.e. (\displaystyle\int_a^b{y} dx ).

The width of each strip ((h)) is calculated by (h = \dfrac{b-a}{n}).

The height of the boundary of each strip can be calculated by working out (y₀ = f(a)), (y_1 = f(a+h)), (y_2 = f(a+2h)), etc

The area of one strip can be calculated by using the formula for the area of a trapezium, e.g. for the first strip:

(Area = \dfrac{1}{2}h(y_0+y1) )

The area of all the strips (an approximation of the area under the curve) is therefore:

(\displaystyle\int_a^b{y} dx \approx \dfrac{1}{2}h(y_0+y_1) + \dfrac{1}{2}h(y_1+y_2) + ... + \dfrac{1}{2}h(y_+y_n))

Factorising out (\dfrac{1}{2}h) gives:

(\displaystyle\int_a^b{y} dx \approx \dfrac{1}{2}h(y_0 + y_1 + y_1 + y_2 + y_2 + ... + y_ + y_ + y_n))

(\implies \boxed{\displaystyle\int_a^b{y} dx \approx \dfrac{1}{2}h(y_0 + 2(y_1 + y_2 + ... + y_) + y_n)} )

Tip: This formula is provided in the formula book but you need to know how to use it.

[b]uThe trapezium rule[/u]/b

(\displaystyle\int_a^b{y} dx \approx \dfrac{1}{2}h(y_0 + 2(y_1 + y_2 + ... + y_) + y_n) )

where (h = \dfrac{b-a}{n})