Trapezium rule

The trapezium rule states that you can approximate the area under a curve by a trapezium, as shown in figure C.4. The area is therefore calculated like this:

Figure C.4: A straight line is used to approximate the curve f (x) between x₁ and x₂. This forms a trapezium (shaded area) which can be used to calculate approximately the area under the curve between these bounds.

$$\int_{{x_1}}^{{x_2}}$$f (x)dx = h$$\left(\vphantom{ \frac{1}{2} f_1 + \frac{1}{2} f_2}\right.$$$${\frac{{1}}{{2}}}$$f₁ + $${\frac{{1}}{{2}}}$$f₂$$\left.\vphantom{ \frac{1}{2} f_1 + \frac{1}{2} f_2}\right)$$ + O(h³f'')

Where the notation f₁ is short hand for f (x₁).

The error due to the approximation, O(h³f''), indicates that this method is exact for polynomials up to degree 1, i.e. straight lines f (x) = ax + b, because for such functions the second derivative f'' = 0. It is clearly an approximation for higher order polynomials (x²,x³, etc.) and other functions. The error will also be small if h is small.

The area of a trapezium is its average height x its width, i.e. ${\frac{{h}}{{2}}}$(f₁ + f₂).