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 term and yₙ means last term.
- (y₁ + y₃ + ..) means the sum of odd terms.
- (y₂ + y₄ + …) means sum of even terms.
Example: Find the Solution using Simpson’s 1/3 rule.
| x | f(x) |
|---|---|
| 0.0 | 1.0000 |
| 0.1 | 0.9975 |
| 0.2 | 0.9900 |
| 0.3 | 0.9776 |
| 0.4 | 0.8604 |
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.
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 quadratic, but they really are quadratics.