PURE MATHEMATICS - Integration

 

The Trapezium Rule

 

 

Theory & Method

example #1

example #2

 

 

 

 

Theory & Method

 

The Trapezium Rule is a method of finding the approximate value of an integral between two limits.

 

The area involved is divided up into a number of parallel strips of equal width.

 

 

trapezium rule diagram#1

 

 

Each area is considered to be a trapezium(trapezoid).

 

If there are n vertical strips then there are n+1 vertical lines(ordinates) bounding them.

 

The limits of the integral are between a and b, and each vertical line has length y1 y2 y3... yn+1

 

width of each strip

 

 

area of first strip = (width of strip) x (av. length of 1st. and 2nd. verticals)

 

 

                            

 

 

Therefore in terms of the all the vertical strips, while the actual integral is given by:

 

 

 

 

the approx. integral is

 

       = (strip width) x [(av. of 1st. and last y-values) + (the sum of all y values between the                                                                        second and second-last value)]

 

 

 

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Example #1

using a strip width of        

 

evaluate        

 

(answer to 2 d.p.)        

 

 

 

 

 

using approx. integral= (strip width) x [(av. of 1st. and last y-values)

 

+ (the sum of all y values between the second and second-last value)]

 

 

 

 

 

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Example #2

using a strip width of  '1' unit

 

evaluate       

 

                  

 

 

 

         

 

 

 

 

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