以下哪一项等于
∫
π
−π
f（x）dx
，正确到小数3位，其中
f（x）={−7x 如果−π≤x≤0
3⁻³因为x 如果0

Question

以下哪一项等于
∫
π
−π
f（x）dx
，正确到小数3位，其中
f（x）={−7x	如果−π≤x≤0
3⁻³因为x	如果0<x≤π。

BlackTom AI · Accepted Answer

Compute ∫_{-π}^0 (−7x) dx and ∫_{0}^{π} 3 dx directly, then add them.

1. Relevant knowledge points and concepts
- Linearity of the definite integral: ∫[a to b] (f+g) = ∫f + ∫g.
- Power rule for antiderivatives: ∫ x dx = x^2/2.
- Integral of a constant c on [a,b] is c(b−a).
- Numerical evaluation of π and π^2 for decimal rounding.

2. Step-by-step reasoning
The piecewise function gives two integrals to evaluate:
I = ∫_{-π}^{π} f(x) dx = ∫_{-π}^{0} (−7x) dx + ∫_{0}^{π} 3 dx.

3. Calculations with intermediate steps
Left part:
∫_{-π}^{0} (−7x) dx = −7 ∫_{-π}^{0} x dx = −7 [x^2/2]_{-π}^{0}.
Compute the antiderivative at the bounds:
[x^2/2]_{-π}^{0} = (0^2/2) − ((−π)^2/2) = 0 − (π^2/2) = −π^2/2.
Multiply by −7: −7 * (−π^2/2) = 7π^2/2.

Right part:
∫_{0}^{π} 3 dx = 3 [x]_{0}^{π} = 3(π − 0) = 3π.

Sum:
I = 7π^2/2 + 3π (exact form).

Numeric evaluation:
π ≈ 3.141592654, π^2 ≈ 9.869604401.
7π^2/2 = 7 * 9.869604401 / 2 ≈ 34.5436154035.
3π ≈ 9.4247779608.
I ≈ 34.5436154035 + 9.4247779608 = 43.9683933643.
Rounded to three decimal places gives 43.968.

4. Why the correct answer is correct
The computation directly follows integration rules; the left integral yields 7π^2/2 and the right yields 3π, whose sum numerically equals 43.968 (to three decimals).

5. Why other options are incorrect
No alternative choices were provided here; any other numeric value would not match the exact expression 7π^2/2 + 3π or its evaluated decimal 43.968.

6. Conclusion
The integral equals 7π^2/2 + 3π, which evaluates to 43.968 (to three decimal places), confirming the provided answer 43.968.

以下哪一项等于 ∫ π −π f（x）dx ，正确到小数3位，其中 f（x）={−7x 如果−π≤x≤0 3⁻³因为x 如果0<x≤π。Single choice

Similar Questions

Find the value of [math: ∫04(−x+3x)dx]

The exact value of [math: ∫−22(4e2x−2e−2x)dx] is:

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

Which of the following is equal to ∫ − 𝜋 𝜋 𝑓 ( 𝑥 ) 𝑑 𝑥 correct to 3 decimal places, where 𝑓 ( 𝑥 ) = { 12 𝑥 if − 𝜋 ≤ 𝑥 ≤ 0 3 − 3 cos ⁡ 𝑥 if 0 < 𝑥 ≤ 𝜋 .

Consider the rational function 𝑓 ( 𝑥 ) = (5 𝑥 +2)/( 𝑥 2 +1) Find ∫ 0 1 𝑓 ( 𝑥 ) 𝑑 𝑥 Enter your answer to 4 decimal places.

Evaluate .

The velocity 𝑣 of a particle at time 𝑡 (in seconds) is measured and the graph 𝑦 = 𝑣 ( 𝑡 ) is shown below. What is the particle's displacement after the first 6 seconds?

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler