Chapter 6: Approximate Integration

Question 1

What is Simpson’s rule? Also, define ∆x and n.

Answer 1

∫^b_a f(x) dx =

(∆x/3) [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + 2f(x_n-2) + 4f(x_n-1) + f(x_n)]

where

∆x = (b-a)/n

and

n is even

Question 2

What is the trapezoidal rule? Also, define ∆x and x_i.

Answer 2

∫^b_a f(x) dx = (∆x/2) [f(x₀) + 2f(x₁) + 2f(x₂) + … + 2f(x_n-1) + f(x_n)]

where

∆x = (b-a)/n

and

x_i = a + i∆x

Question 3

What is the midpoint rule? Also, define ∆x and x_i.

Answer 3

∫^b_a f(x) dx = ∆x [f(x₁) + f(x₂) + … + f(x_n)]

where

∆x = (b-a)/n

and

x_i = .5(x_i-1+x_i) or the midpoint of the two