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Net change and accumulation from a rate

Problem 4.344 · easy

A particle starts at position 500 m with velocity \( \displaystyle v(t) = 2 t + 5 \) m/s. Where is it at \( \displaystyle t = 3 \)?

The net change over [0, T] is the integral of the rate (the Fundamental Theorem of Calculus).
Reviewed
\[ \int\limits_{0}^{3} \left(2 t + 5\right)\, dt = 24 \]
The change over the interval.✓ Proved
\[ 524 \]
Add the starting amount.✓ Proved

Answer \( 524 \approx 524\ \text{m} \)

✓ 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.

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	numerical quadrature of the rate

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — 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.

Every verdict on record (4)

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
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 arithmetic 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.