Integrate the following data using the composite Simpson's 1/3 rule. What is the output? [table]
𝑥x | -5 | -4.5 | -4 | -3.5 | -3 𝑓(𝑥)f(x) | 8 | 5 | 6 | 5 | 7
[/table]

Question

BlackTom AI · Accepted Answer

Using composite Simpson's 1/3 with h = 0.5, apply the formula for n = 4 (even): I ≈ (h/3)[f0 + fn + 4(sum of odd-indexed f) + 2(sum of even-indexed f)].

1) Relevant concepts:
- Composite Simpson's 1/3 rule for equally spaced data with an even number of subintervals: I ≈ (h/3)[f(x0)+f(xn)+4Σf(odd indices)+2Σf(even indices)].
- Here the x-values are equally spaced by h = -4.5 - (-5) = 0.5 (same for each step), and there are 4 subintervals (n = 4), so Simpson's rule applies.

2) Step-by-step computation:
- Label the function values:
  f0 = f(-5) = 8
  f1 = f(-4.5) = 5
  f2 = f(-4) = 6
  f3 = f(-3.5) = 5
  f4 = f(-3) = 7
- Compute h = 0.5.
- Sum odd-indexed values: f1 + f3 = 5 + 5 = 10.
- Sum even-indexed (interior) values: f2 = 6.
- Plug into Simpson's formula:
  I ≈ (h/3)[f0 + f4 + 4(f1+f3) + 2(f2)]
  I ≈ (0.5/3)[8 + 7 + 4(10) + 2(6)]

3) Intermediate arithmetic:
- 8 + 7 = 15
- 4(10) = 40
- 2(6) = 12
- Total inside brackets: 15 + 40 + 12 = 67
- Multiply by h/3: (0.5/3) * 67 = (1/6) * 67 = 67/6 ≈ 11.1666667

4) Why the correct answer is correct:
- This result directly follows from the Simpson composite formula applied to the given equally spaced data. The exact arithmetic gives 67/6 ≈ 11.1667, which rounds to 11.17.

5) Why other options are incorrect:
- 14.57, 8.5, and 13.23 do not match the computed Simpson value 67/6 ≈ 11.1667; they would correspond to different weights or incorrect spacing/sums. The correct weighted combination of endpoints and interior points yields 11.17.

Conclusion: The integral approximated by composite Simpson's 1/3 rule is 67/6 ≈ 11.17, confirming answer a. 11.17.

Integrate the following data using the composite Simpson's 1/3 rule. What is the output? [table] 𝑥x | -5 | -4.5 | -4 | -3.5 | -3 𝑓(𝑥)f(x) | 8 | 5 | 6 | 5 | 7 [/table]单项选择题

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