What is the approximate value of the following integral using a single application of Simpson’s 1/3 rule?[math: f(x)∫09(x3−8x2)dx]

Question

BlackTom AI · Accepted Answer

Simpson’s 1/3 rule with a single application uses n = 2 subintervals, so h = (b − a)/2 = (9 − 0)/2 = 4.5. Simpson’s formula: integral ≈ (h/3)[f(x0) + 4f(x1) + f(x2)].

1) Relevant concepts
- Simpson’s 1/3 rule for n = 2 approximates ∫_a^b f(x) dx by (h/3)[f(a)+4f(a+h)+f(b)] with h = (b−a)/2.
- Simpson’s rule is exact for any polynomial up to degree 3, so it will produce the exact value here because the integrand is a cubic x^3 − 8x^2.

2) Step-by-step calculation
- Endpoints: x0 = 0, x1 = 0 + h = 4.5, x2 = 9.
- Evaluate f(x) = x^3 − 8x^2 at these points:
  f(0) = 0^3 − 8·0^2 = 0.
  f(4.5): 4.5^2 = 20.25, 4.5^3 = 91.125 → f(4.5) = 91.125 − 8·20.25 = 91.125 − 162 = −70.875.
  f(9): 9^2 = 81, 9^3 = 729 → f(9) = 729 − 8·81 = 729 − 648 = 81.
- Apply Simpson’s formula:
  (h/3)[f(0) + 4f(4.5) + f(9)] = (4.5/3)[0 + 4(−70.875) + 81].
  Compute inside: 4(−70.875) = −283.5. Sum: 0 − 283.5 + 81 = −202.5.
  Multiply: (4.5/3) = 1.5, so 1.5 × (−202.5) = −303.75.

3) Verification by exact integration (optional but instructive)
- Antiderivative: ∫(x^3 − 8x^2) dx = x^4/4 − (8/3)x^3.
- Evaluate from 0 to 9: (9^4)/4 − (8/3)·9^3 = 6561/4 − (8·729)/3 = 1640.25 − 1944 = −303.75.
- This matches the Simpson result because Simpson’s rule is exact for cubics.

4) Why the correct answer is correct
- The computed Simpson value is −303.75 and the exact integral is also −303.75, so choice b is correct.

5) Why other options are incorrect
- a (−288.4741), c (−312.8136), and d (−343.5000) do not equal the computed value −303.75 and therefore are incorrect; they likely come from other approximation mistakes or different h/subinterval choices.

Conclusion: The Simpson’s 1/3 single-application approximation (and the exact integral) equals −303.7500, so the correct answer is b.

What is the approximate value of the following integral using a single application of Simpson’s 1/3 rule?[math: f(x)∫09(x3−8x2)dx]单项选择题

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