Approximate Integration- Trapezoidal and Simpson's Rule

In order to evaluate , the interval [a,b] is subdivided into n subintervals

each of length:

x₀ = a
x₁ = a + h
x₂= a + 2h
x₃ = a + 3h

x_i = a + ih
x_n = a + nh = b

Then find the corresponding y = f(x) and use one of the following:

Trapezoidal Rule:

Value of Area; A_T = h / 2 (y₀ + 2y₁ + 2y₂ + . . . + 2y_n-1 + y_n)

Error; E_T = -( h² / 12 ) (b - a) f '' (c); where a £ c £ b.

 

Simpson's Rule:

Value of area; A_s= h / 3 ( y₀ + 4y₁ + 2y₂ + 4y₃ + 2y₄ + . . . + 2y_n-2 + 4y_n-1 + y_n )

Error; E_s = - (h⁴ / 180) ( b - a) f⁽⁴⁾(c); where a £ c £ b.