1. Do Problems 3–6 in this problem set.

2. Modify the Maple code for Simpson's–1/3 rule (see the previous assignment) to turn it into Simpson's–3/8 rule.
   Then use it to approximate the sample problem (integral of sin x on the interval [0, π]) using 9 subintervals.
   (Also see the pseudocode for Simpson's–3/8 rule.)

3. Now re-run your Simpson's–3/8 rule code using 18 subintervals.

4. Use the Simpson's–3/8 rule code to approximate the integral of   e^–x2  on the interval [0,3] using:
   1. 9 subintervals        Answer:   0.88619 38607 33682 32028
   2. 18 subintervals      Answer:   0.88620 71620 01857 85809
   3. Use Richardson extrapolation to obtain an improved estimate.
      Answer:   0.88620 80487 53069 56062
   4. Without knowing the actual errors, determine how much more accurate result (b) should be than result (a).

5. Play around with the code to approximate integrals of other functions on different intervals.
   

6. In class I derived the formula for the trapezoidal rule. Now you will derive the composite formula for Simpson's–3/8 rule:
   1. First construct the Newton-Gregory interpolating polynomial P_[0–3](x) containing points [0,1,2,3]. It approximates function f(x) on Section 1.

   2. Integrate the polynomial on Section 1: [x₀, x₃] and simplify the result.
      You should obtain    I₁ = (3/8) · h · ( f₀ + 3 f₁ + 3 f₂ + f₃ ).

   3. Use the formula for I₁ on Section 1 as the template to construct the formula for I₂ on Section 2, I₃ on Section 3, etc.

   4. Add I₁ + I₂ + I₃ + · · ·   to obtain the composite Simpson's–3/8 rule.

Newton-Cotes Methods:   All the integration methods we've studied so far — the trapezoidal rule, Simpson's–1/3 rule, and Simpson's–3/8 rule — belong to a group of methods call Newton–Cotes methods. Newton–Cotes methods are based on Newton–Gregory interpolating polynomials (so they require equally spaced abscissas).
See a summary of some of the Newton–Cotes integration rules.

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