Part A

The trapezium and Simpson’s rules are examples of quadrature rules.  A quadrature rule seeks to approximate an integral as a weighted sum of function values

\ab f(x) dx ≈ wk f(xk ),

where the xk  are the quadrature nodes and the wk  are called the quadrature weights. A quadrature rule is exact to order k, if it evaluates polynomials up to degree k exactly. A Newton-Cotes scheme of order n uses the Lagrange basis functions to construct

the interpolation weights. Given nodes xk  = a+kh, 0 ≤ k ≤ n, where h = (b− a)/n,

these points. If

n

pn  = 之 Lk f(xk ),

k=0

then

\ab f(x) dx ≈ In (f) := \ab pn (x) dx = wk f(xk ),

where wk  = la(b) Lk (x) dx.

(4.1)    With the usual notation for Newton-Cotes quadrature and using equally spaced

quadrature points xk   = a + kh for h =  (b − a)/n and k = 0, 1, . . . ,n, show that

By considering the polynomial [x − (a + b)/2]n+1, show that the Newton-Cotes

whenever n is even.

(4.2)    Consider a function f(x) that is known at points xi to accuracy ϵi . Analyse the effects of data errors ϵi for both Simpson’s rule and the formula

\0 1 f(x) dx ≈  (2f() − f() + 2f())

by estimating |I(f) − I(fϵ )| where I(f), I(fϵ ) are the results of the integration rule

in quadrature formula?

Part B

(4.3)    Approximate the following integrals using the Trapezium and the Simpson rule: