## 9.4 The trapezium rule

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

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

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 $$y_0 = f(a)$$, $$y_1 = f(a+h)$$, $$y_2 = f(a+2h)$$, 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:

$$Area = \dfrac{1}{2}h(y_0+y_1)$$

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

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

Factorising out $$\dfrac{1}{2}h$$ gives:

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

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

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

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

where $$h = \dfrac{b-a}{n}$$
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