1. 3. Numerical integration (Numerical quadrature) .
Given the continuous function f(x) on [a,b], approximate
Newton-Cotes Formulas:
For the given abscissas, approximate the integral I(f) by the integral of interpolating formula with degree n, I(pn) .
Formulas that use end points a, and b as data points are called closed formulas.
Those not use the end points called open (semi-open) formulas.

2. Newton-Cotes formula for with equally spaced abscissas .
Rectangle rule
Mid point rule.
Trapezoidal rule
Simpson’s rule
Simpson’s 3/8 rule
Bode’s rule
Rectangle and Mid point rules are the (semi-)open formulas.
Definition: Degree of Precision (or Accuracy) of a quadrature rule In(f) is the positive integer D, if
I(xk) = In(xk) for the degree k · D, and
I(xk) ¹ In(xk) for the degree k = D + 1.

3. Closed Newton-Cotes formula of degree D=n=8.
Weights wi contain some negative coefficients.
This thorem suggests that the higher order Newton-Cotes formula wouldn’t be useful for practical numerical computations.
Piecewise ! composite rules.
Optimize data points ! A family of Gauss formulas.

4. Theorem: (The error associated with Newton-Cotes formulas.)
For Newton-Cotes formula with n+1 abscissas
(open or closed)
a) For even n, and f(x)2 C(n+2)(a,b), 9 x2(a,b) such that,
b) For odd n, and f(x)2 C(n+1)(a,b), 9 x2(a,b) such that,
n=even cases are generally better in the degree of precision.
Constants c, c’ depend on n and type of formula open or closed.
For a given n, c of the closed formulas are typically smaller than the open formula. The closed formulas are more used in practice.
If the function has a singularity at the end point, open formulas can be useful.

5. Theorem: (Weighted Mean-Value Theorem for Integrals.)
Exc. 3-1) Prove this.
The above theorem is used to determine the error of a Newton_Cotes formula.
Ex) Trapezoidal formula.

6. Exc. 3-2) Verify Boole’s rule using an algebraic computing software.
Exc. 3-3) Derive the error term for the Simplson’s rule using the interpolation error formula,
Exc 3-4) Derive the error term for the mid-point rule and Simpson’s rule.
Exc 3-5) Derive the error associated with Newton-Cotes formulas.

7. Extended (composite, compound) formula.
Trapezoidal formula
Simpson formula

8. For the composite trapezoidal rule,
Theorem: (Euler-Maclaurin Sum Formula).
If the f(x) has odd derivatives that are equal at the end points of interval [a,b], such as a periodic function on [a,b], the composite trapezoidal rule becomes more accurate. (Also extended mid-point rule.)

9. Romberg integration.
Extrapolation applied to the composite trapezoidal rule.
Euler-Maclaurin summation formula,
Romberg approximations is written Rk,j ,
Level of extrapolation 
 Step size
Exc 3-6) Using Romberg integration, calculate the definite integral
and estimate the error up to R4,4 .

10. Gaussian quadratures.
Approximating the integral in the form,
optimize the location of data points and the associated weights,
in the way that the integrals of polynomials have exact value,
(From 2 n parameters wi and xi , a quadrature formula with the degree of precision 2n-1 can be constructed at best.)

11. Exc 3-7) Prove the above theorem.

12. Exc 3-8) Prove the above theorem.
hint) show the following facts.

13. Exc 3-9) Try some of the above.
More topics for the numerical integration.
Higher precision integration formulas.
ex) IMT type formula, (DE formula.)
Integration of improper integrals.
(i.e. infinite integral region, or discontinuity of integrand.)
Integration of multivariate functions.
ex) Monte-Carlo. Multivariate Gauss formulas.