Apply a single application of Simpson’s 3/8 rule to evaluate the following integral. What is the output?[math: f(x)∫911(cos⁡(5x)−7)dx]

Question

BlackTom AI · Accepted Answer

Simpson’s 3/8 rule for a single application on [a,b] with three subintervals (n=3) is
Integral ≈ (3h/8)[f(x0) + 3f(x1) + 3f(x2) + f(x3)],
where h = (b−a)/3 and xk = a + k h.

1) Relevant concepts
- Use Simpson’s 3/8 rule with a = 9, b = 11 (so three equal subintervals).
- f(x) = cos(5x) − 7; evaluate f at the four nodes and apply the formula.

2) Step-by-step reasoning and intermediate calculations
- h = (11 − 9)/3 = 2/3.
- Nodes: x0 = 9, x1 = 9 + 2/3 = 9.6666667, x2 = 9 + 4/3 = 10.3333333, x3 = 11.
- Compute 5x values and cosines (radians):
  5x0 = 45 → cos(45) ≈ 0.52529377 → f0 = 0.52529377 − 7 = −6.47470623
  5x1 = 48.3333333 → cos(48.3333333) ≈ −0.35343124 → f1 = −0.35343124 − 7 = −7.35343124
  5x2 = 51.6666667 → cos(51.6666667) ≈ 0.16879900 → f2 = 0.16879900 − 7 = −6.83120100
  5x3 = 55 → cos(55) ≈ 0.02212695 → f3 = 0.02212695 − 7 = −6.97787305
- Apply Simpson’s 3/8 factor: 3h/8 = 3*(2/3)/8 = 2/8 = 1/4.
- Weighted sum: S = f0 + 3f1 + 3f2 + f3
  = −6.47470623 + 3(−7.35343124) + 3(−6.83120100) + (−6.97787305)
  = −56.00647600 (sum of terms).
- Multiply by 1/4: Integral ≈ 0.25 * (−56.00647600) = −14.001619 ≈ −14.0017.

3) Why this is correct
- The formula was applied correctly with h = 2/3 and the four node evaluations; arithmetic yields −14.001619 which rounds to −14.0017.

4) Why the other options are incorrect
- Option a (−28.0837) and c (−20.0497) and d (−7.2828) do not match the correctly computed weighted sum times 1/4. They would require different node values or an incorrect application of the rule (wrong h, wrong weights, or arithmetic error).

Conclusion: The Simpson’s 3/8 single-application approximation is −14.0017, confirming answer b.

Apply a single application of Simpson’s 3/8 rule to evaluate the following integral. What is the output?[math: f(x)∫911(cos⁡(5x)−7)dx]单项选择题

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