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# Difference between Simpson ‘s 1/3 rule and 3/8 rule

In Simpson’s 1/3 rule, we approximate the polynomial based on quadratic approximation. In this, each approximation actually covers two of the subintervals. This is why we require the number of subintervals to be even. Some of the approximations look more like a line than a quadric, but they really are quadratics.

Formula of Simpson’s¹/₃ rule

ₐ∫ᵇ f (x) dx = h/₃ [(y₀ + yₙ) + 4 (y₁ + y₃ + ..) + 2(y₂ + y₄ + ..)]

where,

a, b is the interval of integration

h = (b – a)/ n

y₀ means the first terms and yₙ means last terms.

(y₁ + y₃ + ..) means the sum of odd terms.

(y₂ + y₄ + …) means sum of even terms.

Example: Find Solution using Simpson’s 1/3 rule.

Solution:

Using Simpson’s 1/3 rule

ₐ∫ᵇ f (x) dx = h/₃ [(y₀ + yₙ) + 4 (y₁ + y₃ + …) + 2 (y₂ + y₄ + …)]

h = 0.1

ₐ∫ᵇ f (x) dx = 0.1/3 [(1+0.8604)+4×(0.9975+0.9776)+2×(0.99)]

ₐ∫ᵇ f (x) dx = 0.1/3 [(1+0.8604)+4×(1.9751)+2×(0.99)]

ₐ∫ᵇ f (x) dx = 0.39136

Solution of Simpson’s 1/3 rule =  0.39136 In Simpson’s 3/8 rule, we approximate the polynomial based on quadratic approximation. However, each approximation actually covers three of the subintervals instead of two.

Formula of Simpson’s 3/8 rule

ₐ∫ᵇ f (x) dx = 3h/8[(y₀ + yₙ) + 3(y₁ + y₂ + y₄ + …) + 2(y₃ + y₆ +…)]

where,

a, b is the interval of integration

h = (b – a )/ n

y₀ means the first terms and yₙ means the last terms.

( y₁ + y₂ + y₄ + … ) means the sum of remaining terms.

( y₃ + y₆ +…) means the multiples of 3.

Example: Find a Solution using Simpson’s 1/3 rule.

Solution:

Using Simpson’s 3/8 rule:

ₐ∫ᵇ f (x) dx = 3h/8[(y₀ + yₙ) + 3(y₁ + y₂ + y₄ + …) + 2(y₃ + y₆ +…)]

h = 0.1

ₐ∫ᵇ f (x) dx = 3h/8 [(y0+y4)+2(y3)+3(y1+y2)]

ₐ∫ᵇ f (x) dx = 3* 0.1/8 [(1+0.8604)+2×(0.9776)+3×(0.9975+0.99)]

ₐ∫ᵇ f (x) dx = 3* 0.1/8 [(1+0.8604)+2×(0.9776)+3×(1.9875)]

ₐ∫ᵇ f (x) dx = 0.36668

Solution of Simpson’s 3/8 rule = 0.36668

Following is a table of differences between the Simpson’s 1/3 rule and Simpson’s 3/8 rule

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