Home›Calculus 3›Line integrals of scalar functions›Problem 12.412

Line integrals of scalar functions

Problem 12.412 · medium

Evaluate \( \displaystyle \int_C x + y + z \, ds \), where \( \displaystyle C \) is the quarter circle of radius 3 in the xy-plane, from (3, 0, 0) to (0, 3, 0).

Parametrize: r(t) = <3*cos(t), 3*sin(t), 0>, 0 ≤ t ≤ pi/2; ds = ‖r′(t)‖ dt.
Reviewed
\[ \sqrt{\left(\frac{d}{d t} 0\right)^{2} + \left(\frac{d}{d t} 3 \sin{\left(t \right)}\right)^{2} + \left(\frac{d}{d t} 3 \cos{\left(t \right)}\right)^{2}} = 3 \]
‖r′(t)‖.✓ Proved
\[ \int\limits_{0}^{\frac{\pi}{2}} \left. x + y + z \right|_{\substack{ x=3 \cos{\left(t \right)}\\ y=3 \sin{\left(t \right)}\\ z=0 }} \cdot 3\, dt = 18 \]
∫ f(r(t)) ‖r′(t)‖ dt.✓ Proved

Answer \( 18 \)

✓ 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — The parametrization correctly models the quarter circle in the xy-plane from (3,0,0) to (0,3,0). The magnitude of the derivative is correctly calculated as 3, and the resulting integral evaluates to 18.
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 2026-10-18 with SymPy 1.14.0.