1. Evaluate the integrand function f at the points x_i

2. Determine the polynomial of degree n-1 that interpolations the function values at those points

3. Take the integral of the interpolant as an approximation to the integral of the original function

Method of undetermined coef with monomial basis

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• Choose equally spaced nodes in the interval [a,b].  
• An n-point open Newton-Cotes rule has nodes x_i=a + i(b-a)/(n+1)
• An n-point closed Newton-Cotes rule has nodes x_i=a+(i-1)(b-a)/(n-1)
• An n-point rule with odd n has dress one greater than that of the polynomial on which it is based

Interpolate function value at the midpoint of the interval by polynomial of degree 0.

M(f) = (b-a) f((a+b)/2)

Interpolate function values at the two endpoints of the interval by a polynomial of degree one (straight line).

T(f) = (b-a)/2 (f(a)+f(b))

Interpolate function values at two endpoints and midpoint by polynomial of degree two.

S(f) = (b-a)/6 (f(a) +4f((a+b)/2) + f(b))

• On a given interval [a,b], subdivide the interval into k subintervals, typically of uniform length h = (b-a)/k.
• Apply n-point simple quadrature rule in each subinterval.
• Take the sum of these results as approximate value of the integral.

• Take a pair of quadrature rules whose difference provides an error estimate, or a single rule at two different levels of subdivision
• Apply both rules on the initial interval of integration.  If the resulting approximate values differ by more than tolerance, divide interval into two or more subintervals and repeat

• Compute approximate values for some step sizes and then estimate what value would be for step size zero.
• F(h) - value for some step size h
• a₀- approximate solution to F(0)
• a₀ = F(h) + (F(h) - F(h/q))/(q^-p-1) + O(h^r)

• Richardson extrapolation with composite trapezoid rule
• F(h) = a₀+a₁h²+O(h⁴)
• F(0) = a₀=F(h) + (F(h) - F(h/2))/(2^-2-1)
• F(0) = (4 F(h/2) - F(h)) / 3

Inherently Sensitive

Small perturbations in the data can cause large changes in the result