Calculus of vector-valued functions
Problem 9.253 · medium
Evaluate \( \displaystyle \int_0^{\pi} \left\langle t^{2}, 2 t, \cos{\left(t \right)} \right\rangle\, dt \).
- Integrate each component separately.Reviewed
- \[ \left[\begin{matrix}\int\limits_{0}^{\pi} t^{2}\, dt\\\int\limits_{0}^{\pi} 2 t\, dt\\\int\limits_{0}^{\pi} \cos{\left(t \right)}\, dt\end{matrix}\right] = \left[\begin{matrix}\frac{\pi^{3}}{3}\\\pi^{2}\\0\end{matrix}\right] \]Component by component.✓ Proved
Answer \( \left\langle \frac{\pi^{3}}{3}, \pi^{2}, 0 \right\rangle \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | each component by numerical quadrature |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the component-wise integration rule for vector-valued functions. The setup and execution are sound.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the component-wise integration rule for vector-valued functions. The setup and execution are sound.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the linearity of integration to vector-valued functions by integrating each component separately. The resulting integrals and their evaluations are correct.gpt-oss:20b: pass 2026-10-05
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/vector_function_calculus, checked 2026-10-05 with SymPy 1.14.0.