Section 6.4: Other Integration Methods

Trapezoidal Rule

Let f be a function defined on an interval [a, b]. Partition [a, b] into n subintervals of equal length Δx = (b - a)/n with endpoints:

a = x₀ < x₁ < x₂ < ··· < x_n = b

Then:

T_n = [f(x₀) + 2f(x₁) + 2f(x₂) + ··· + 2f(x_{n - 1}) + f(x_n)]Δx/2

is an approximation of ∫ab f(x) dx.

Simpson's Rule

Let f be a function defined on an interval [a, b]. Partition [a, b] into 2n subintervals of equal length Δx = (b - a)/2n with endpoints:

a = x₀ < x₁ < x₂ < ··· < x_2n = b

Then:

S_2n = [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + 2f(x₄) + ··· + 4f(x_{2n - 1}) + f(x_2n)]Δx/3

is an approximation of ∫ab f(x) dx.