Areas
 Trapezoidal Rule
 Simpson's Rule
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 Example

Simpson's Rule

The Trapezoidal Rule can be improved by assuming that each two adjacent sub-areas are a single bounded parabola rather than each sub-area being a trapezoid.

For the area contained between 01 and 03; A = Trapezoid (abdea) + parabolic area (agefa) A = (01 + 03)L + 2/3(area bounded by parabola) A = (01 + 03)L + 2/3 x 2L[02 - (01 + 03)/2] A = L[01 + 402 + 03]/03 and this may be repeated and summed for a total area of an irregular figure provided that the number of offsets is odd.

This assumption leads to Simpson's Rule for irregular areas and is quoted as follows;

A = [(O₁ + O_n) + 2(O₃ + O₅ + O_n-2) + 4(O₂ + O₄ + O_n-1)]

A = [S(1st + last offset) + 2S(odd offsets) + 4S(even offsets)]

This formula is more accurate but has the disadvantage that n must be odd. In this case it is not possible to directly compute the total area AXYZCBA. Instead the area AXYBA is computed using Simpson's Rule and the additional area BYZCB must be computed separately. This could have been avoided if the irregular area had been originally subdivided into an odd number of sub-areas.