Home›Calculus 2›Area and surface area with parametric curves›Problem 8.255

Area and surface area with parametric curves

Problem 8.255 · medium

Find the area under the curve \( \displaystyle x = t^{2},\ y = t^{3} \) for \( \displaystyle 0 \le t \le 1 \) (above the \( \displaystyle x \)-axis).

x increases with t here, so A = ∫ y dx = ∫ y(t) x′(t) dt.
Reviewed
\[ t^{3} \frac{d}{d t} t^{2} = 2 t^{4} \]
y·x′(t).✓ Proved
\[ \int\limits_{0}^{1} 2 t^{4}\, dt = \frac{2}{5} \]
Integrate.✓ Proved

Answer \( \frac{2}{5} \)

✓ 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	the curve as a 20,000-segment polyline: trapezoids under it, or the frusta it sweeps

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the formula for the area under a parametric curve, verifies the monotonicity of x to ensure the standard integral applies, and performs the integration correctly.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the formula for the area under a parametric curve, verifies the monotonicity of x to ensure the standard integral applies, and performs the integration correctly.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly sets up the integral for the area under a parametric curve and computes it accurately.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly sets up the integral for the area under a parametric curve and computes it accurately.
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/parametric_area_surface, checked 2026-10-17 with SymPy 1.14.0.