Volumes with known cross sections
Problem 5.393 · medium
The base of a solid is the region under \( \displaystyle y = \sqrt{x} \) for \( \displaystyle 0 \le x \le 16 \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are semicircles. Find the volume of the solid.
- \[ \sqrt{x} \]At position x the base of the cross-section has length (top − bottom).✓ Proved
- \[ \frac{\pi x}{8} \]Area of one cross-section: pi/8·(side)².✓ Proved
- \[ \int\limits_{0}^{16} \frac{\pi x}{8}\, dx = 16 \pi \]Volume = ∫ A(x) dx across the base.✓ Proved
Answer \( 16 \pi \)
Every line of this solution was proved by the computer algebra system SymPy. 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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 cross-sectional area |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the side length of the semicircle as x instead of 2*sqrt(x). Since the cross-sections are perpendicular to the x
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the side length of the semicircle as x instead of 2*sqrt(x). Since the cross-sections are perpendicular to the xgpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: inconclusive 2026-10-09 — reviewer returned a non-object
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/volume_cross_sections, checked 2026-10-09 with SymPy 1.14.0.