# Trapezoidal Rule

The trapezoidal rule uses a trapezium for the approximate area under a curve. A trapezium generally gives a better approximation to the area than a rectangle.

### Examples

1. Find an approximation for $\int _{ 0 }^{ 1 }{ { x }^{ 3 } } dx$ using the trapezoidal rule with 2 sub-intervals.

2. Find an approximation for $\int _{ 2 }^{ 3 }{ \cfrac { 2 }{ x-1 } } dx$, using the trapezoidal rule with 4 sub-intervals, correct to 3 decimal places.

When surveyors need to find the area of an irregular piece of land, they measure regular strips and use an approximation method such as the trapezoidal rule.

## Simpson's Rule

Another more generally accurate version of trapezoidal rule, since it makes use of parabolic arcs instead of straight lines.

The general formula for n equal sub-intervals are given below:

### Examples

1. Use Simpson's rule with 5 function values to find an approximation for $\int _{ 0 }^{ 2 }{ \cfrac { dx }{ x+1 } }$.

2. Use Simpson's rule with 9 function values to find an approximation for $\int _{ 1 }^{ 5 }{ \cfrac { dx }{ { x }^{ 2 } } }$.