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 } }  } .