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Net change and accumulation from a rate

Problem 4.349 · easy

A particle starts at position 50 m with velocity \( \displaystyle v(t) = 2 t + 5 \) m/s. Where is it at \( \displaystyle t = 4 \)?

The net change over [0, T] is the integral of the rate (the Fundamental Theorem of Calculus).
\[ \int\limits_{0}^{4} \left(2 t + 5\right)\, dt = 36 \]
The change over the interval.✓ Proved
\[ 86 \]
Add the starting amount.✓ Proved

Answer \( 86 \approx 86\ \text{m} \)

Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	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	numerical quadrature of the rate

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The integral is computed from t=0 to t=4, but the particle starts at t=0 with position 50m. The problem implies the motion starts at t=0 (or the ini

Every verdict on record (4)

qwen3.6:27b-mlx: inconclusive 2026-10-04 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The integral is computed from t=0 to t=4, but the particle starts at t=0 with position 50m. The problem implies the motion starts at t=0 (or the ini
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the Fundamental Theorem of Calculus to find the displacement and adds it to the initial position. The logic is sound and the result is correct.
gpt-oss:20b: pass 2026-10-04

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/net_change, checked 2026-10-04 with SymPy 1.14.0.