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Calculus of vector-valued functions

Problem 9.338 · medium

Evaluate \( \displaystyle \int_0^{2} \left\langle \sqrt{t}, t^{3}, e^{2 t} \right\rangle\, dt \).

Integrate each component separately.
Reviewed
\[ \left[\begin{matrix}\int\limits_{0}^{2} \sqrt{t}\, dt\\\int\limits_{0}^{2} t^{3}\, dt\\\int\limits_{0}^{2} e^{2 t}\, dt\end{matrix}\right] = \left[\begin{matrix}\frac{4 \sqrt{2}}{3}\\4\\- \frac{1}{2} + \frac{e^{4}}{2}\end{matrix}\right] \]
Component by component.✓ Proved

Answer \( \left\langle \frac{4 \sqrt{2}}{3}, 4, - \frac{1}{2} + \frac{e^{4}}{2} \right\rangle \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the definition of the integral of a vector-valued function by integrating each component separately. The algebraic results are verified.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the definition of the integral of a vector-valued function by integrating each component separately. The algebraic results are verified.
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the linearity of the integral to vector-valued functions by integrating each component separately. The bounds and integrands are handled correctly.
gpt-oss:20b: pass 2026-10-07

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 2026-10-07 with SymPy 1.14.0.