Volumes with known cross sections
Problem 5.253 · medium
The base of a solid is the disk \( \displaystyle x^2 + y^2 \le 4 \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are equilateral triangles. Find the volume of the solid.
- \[ 2 \sqrt{4 - x^{2}} \]At position x the base of the cross-section has length (top − bottom).✓ Proved
- \[ \frac{\sqrt{3} \left(16 - 4 x^{2}\right)}{4} = - \sqrt{3} x^{2} + 4 \sqrt{3} \]Area of one cross-section: sqrt(3)/4·(side)².✓ Proved
- \[ \int\limits_{-2}^{2} \left(- \sqrt{3} x^{2} + 4 \sqrt{3}\right)\, dx = \frac{32 \sqrt{3}}{3} \]Volume = ∫ A(x) dx across the base.✓ Proved
Answer \( \frac{32 \sqrt{3}}{3} \)
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: pass — The solution correctly identifies the side length of the equilateral triangles as the chord length of the disk, calculates the area correctly, and integrates over the correct bounds.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the side length of the equilateral triangles as the chord length of the disk, calculates the area correctly, and integrates over the correct bounds.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.