## Trapezium Method for Approximating the Area Under a Curve

The trapezium method allows you to approximate the area under a curve by breaking the curve up into and number of trapeziums whose areas can be easily calculated and then adding these areas up. This can be seen below.

When using the trapezium method the widths of the trapeziums used can be different, however it is often easier to calculate the total area (using the formula explained later) if all of the trapeziums are of equal width.

To improve the accuracy of the approximation you can use more trapeziums (the process of integration is simply allowing there to be an infinite number of trapeziums)

The area A of a trapezium with height (width when vertical in the approximation) h, and parallel sides a and b is given by

**A=h(A+B)/2**

To calculate the approximation we can let the x-values where each of the sides of the trapeziums touch the x-axis be x_{0},x_{1},x_{2}… and the y values where they cut the curve be y_{0},y_{1}…

This means the the width of the first trapezium is x_{1}-x_{0} which we will let equal d which is the same for all the trapeziums if we let their widths be equal. The parallel sides are of length y_{0} and y_{1} so the area A_{0} is given by

A_{0} = d(y_{0}+y_{1})/2

The total area A under the curve is found by adding the areas of all of the trapeziums.

A = d(y_{0}+y_{1})/2 + d(y_{1}+y_{2})/2 + d(y_{2}+y_{3})/2 + d(y_{3}+y_{4})/2 +…….

However since d is a common factor it can brought outside in a bracket. Also all the y co-ordinates occur twice (in two consecutive trapeziums) apart from y_{0} and the last y co-ordinate but all the terms are also halved this means that all but the first and last y values should be counted once and the first and last halved so we get the trapezium rule as follows

**A = d(y _{1}+y_{2}+y_{3} + … + y_{n-1} +(y_{0}+y_{n})/2)**

where there are n trapeziums.

## Area And Circumference of a Circle : pi

**This site is now at www.breakingwave.co.nr**

This is a basic guide to using pi to find the area and circumference of a circle using pi. And also explores why pi makes our formulae work.

area =πr^{2}

circumference = 2πr or πd

where r = radius and d=diameter

**Area**

^{2}. This is simple enough to use, we multiply the radius by itself and then by pi.

**Does this make sense?**

^{2}. Of course the circle’s area is a bit smaller so we need to find the ratio between the areas of the square and circle. If we then times this value by four we have a magic constant to multiply r squared by to find the area of a circle (we times by four because we need the area of 4 quadrants and r squared gives us one).

**Circumference**

**Does this also make sense?**

**Conclusion**

perimeter of square to circumference of circle.

By David Woodford

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