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Simpson's Three-Eighth Rule-The Easy Way

 Simpson's three-eighths rule is a very important rule in numerical integration. In this case, we subdivide [a, b] into a number of subintervals of equal length such that the number of subintervals is divisible by 3. That is, the number of intervals must be 6 or 9 or 12, etc. so that we get 7 or 10 or 13 nodal points, etc. 

Simpson's 3/8th rule has some disadvantages. 

i) The number of subintervals must be divisible by 3.

ii) It is of the same order as Simpson's 1/3rd rule which only requires that the number of nodal points must be odd

iii) The error constant c in the case of Simpson's 3/8th rule is c=3/80, which is much larger than the error constant c=1/90, in the case of Simpson's 1/3rd rule. Therefore, the error in the case of Simpson's 3/8th rule is larger than the error in the case of Simpson's 1/3rd rule. due to these disadvantages, Simpson's 3/8th rule is not used in practice.

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