以下哪一项等于$$\int _{-\pi }^{\pi } f（x） dx$$，正确到小数3位，其中 $$f（x） = \begin{cases} - 7 x & \text {if } -\pi \leq x \leq 0 \ 3 - 3 \cos {x} & \text {if } 0

Question

以下哪一项等于$$\int _{-\pi }^{\pi } f（x） dx$$，正确到小数3位，其中 $$f（x） = \begin{cases} - 7 x & \text {if } -\pi \leq x \leq 0 \ 3 - 3 \cos {x} & \text {if } 0 < x \leq \pi \end{cases} 。$$

BlackTom AI · Accepted Answer

Integrate each piece separately: the integral splits as ∫_{-π}^0 (-7x) dx + ∫_0^{π} (3 - 3 cos x) dx.

Relevant concepts:
- Basic antiderivatives: ∫ x dx = x^2/2, ∫ cos x dx = sin x.
- For a piecewise function, integrate each interval and add results.
- Round the final numeric result to three decimal places.

Step-by-step computation:
1) ∫_{-π}^0 (-7x) dx = -7 ∫_{-π}^0 x dx.
   Compute the inner integral: ∫_{-π}^0 x dx = [x^2/2]_{-π}^0 = 0 - ((-π)^2)/2 = -π^2/2.
   Multiply by -7: -7 * ( -π^2/2 ) = 7π^2/2.

2) ∫_0^{π} (3 - 3 cos x) dx = 3 ∫_0^{π} (1 - cos x) dx
   = 3 [ x - sin x ]_0^{π} = 3 (π - sin π - (0 - sin 0)) = 3π
   since sin π = sin 0 = 0.

3) Sum the two parts:
   Total = 7π^2/2 + 3π.

Numerical evaluation (showing intermediate arithmetic):
- π ≈ 3.141592653589793
- π^2 ≈ 9.869604401089358
- 7π^2/2 = 3.5 × 9.869604401089358 ≈ 34.54361540381275
- 3π ≈ 3 × 3.141592653589793 ≈ 9.42477796076938
- Total ≈ 34.54361540381275 + 9.42477796076938 = 43.96839336458213

Rounded to three decimal places: 43.968.

Why this is correct and common mistakes:
- A common error is to treat -7x as an odd function and assume its integral over symmetric limits is zero; here the function is piecewise and the positive-side integrand is different, so that symmetry argument does not apply.
- Another mistake is mishandling signs when integrating x over [-π,0]; the correct evaluation yields a positive contribution 7π^2/2.
- Mis-evaluating ∫ cos x as -cos x instead of sin x would also give a wrong numeric result.

Conclusion: The integral equals 43.968 (to three decimal places), which matches the provided answer.

以下哪一项等于\[\int _{-\pi }^{\pi } f（x） dx\]，正确到小数3位，其中 \[f（x） = \begin{cases} - 7 x & \text {if } -\pi \leq x \leq 0 \\ 3 - 3 \cos {x} & \text {if } 0 < x \leq \pi \end{cases} 。\]单项选择题

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