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

Question

BlackTom AI · Accepted Answer

Composite trapezoidal rule: ∫_a^b f(x) dx ≈ h [ (1/2)f0 + f1 + f2 + ... + f_{n-1} + (1/2)f_n ].

1. Relevant knowledge points and concepts
- The data are evenly spaced: x = -5, -4.5, -4, -3.5, so the step size h = 0.5.
- Use the composite trapezoidal rule for equally spaced data to approximate the integral over [-5, -3.5].
- Endpoints f0 and f_n receive half weight; interior points receive full weight.

2. Complete reasoning process step by step
- Label values: f0 = f(-5) = 8, f1 = f(-4.5) = 5, f2 = f(-4) = 6, f3 = f(-3.5) = 5.
- Apply the formula with h = 0.5 and n = 3 subintervals.

3. Calculations with intermediate steps
- Compute weighted sum: (1/2)f0 + f1 + f2 + (1/2)f3
  = (1/2)*8 + 5 + 6 + (1/2)*5
  = 4 + 5 + 6 + 2.5
  = 17.5.
- Multiply by h: Integral ≈ h * 17.5 = 0.5 * 17.5 = 8.75.
- (Alternative segment-by-segment check)
  Area_1 = 0.5 * (8 + 5)/2? (correct formula: (b-a)*(f(a)+f(b))/2)
  = 0.5 * (8+5)/2 = 0.5 * 6.5 = 3.25.
  Area_2 = 0.5 * (5+6)/2 = 0.5 * 5.5 = 2.75.
  Area_3 = 0.5 * (6+5)/2 = 2.75.
  Sum = 3.25 + 2.75 + 2.75 = 8.75 (confirms result).

4. Why the correct answer is correct
- The computation directly follows the composite trapezoidal rule with correct h and weights; arithmetic yields 8.75 exactly. The alternative segment calculation (sum of three trapezoids) gives the same value, confirming correctness.

5. Why the other options are incorrect
- b. 11.71, c. 9.51, and d. 13.56 do not match the exact trapezoidal calculation above; they would result from incorrect h, misweighting endpoints, or arithmetic errors. None of them equals the correctly computed 8.75.

6. Conclusion
- The composite trapezoidal approximation is 8.75, so the correct choice is (a) 8.75, matching the provided answer.

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

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