Home›Calculus 3›Line integrals of scalar functions›Problem 12.318

Line integrals of scalar functions

Problem 12.318 · medium

Evaluate \( \displaystyle \int_C z \, ds \), where \( \displaystyle C \) is the segment from (0, 1, 0) to (1, 0, 0).

Parametrize: r(t) = <t, 1 - t, 0>, 0 ≤ t ≤ 1; ds = ‖r′(t)‖ dt.
Reviewed
\[ \sqrt{\left(\frac{d}{d t} 0\right)^{2} + \left(\frac{d}{d t} t\right)^{2} + \left(\frac{d}{d t} \left(1 - t\right)\right)^{2}} = \sqrt{2} \]
‖r′(t)‖.✓ Proved
\[ \int\limits_{0}^{1} \sqrt{2} \left. z \right|_{\substack{ x=t\\ y=1 - t\\ z=0 }}\, dt = 0 \]
∫ f(r(t)) ‖r′(t)‖ dt.✓ Proved

Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

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
3	✓ 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	f at each segment's midpoint times its length, over a 20,000-segment polyline

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The parametrization correctly models the line segment, and the scalar line integral is evaluated correctly, yielding 0.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The parametrization correctly models the line segment, and the scalar line integral is evaluated correctly, yielding 0.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The parametrization is correct for the specified segment, and the integrand z is identically zero on this path, leading to the correct result.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The parametrization correctly models the line segment, and the scalar line integral is evaluated correctly, yielding 0.
gpt-oss:20b: pass 2026-10-14

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/line_integral_scalar, checked 2026-10-17 with SymPy 1.14.0.