Volumes with known cross sections
Problem 5.325 · medium
The base of a solid is the region between \( \displaystyle y = 16 - x^{2} \) and the \( \displaystyle x \)-axis. Cross-sections perpendicular to the \( \displaystyle x \)-axis are isosceles right triangles with a leg in the base. Find the volume of the solid.
- \[ 16 - x^{2} \]At position x the base of the cross-section has length (top − bottom).✓ Proved
- \[ \frac{\left(16 - x^{2}\right)^{2}}{2} = \frac{x^{4}}{2} - 16 x^{2} + 128 \]Area of one cross-section: 1/2·(side)².✓ Proved
- \[ \int\limits_{-4}^{4} \left(\frac{x^{4}}{2} - 16 x^{2} + 128\right)\, dx = \frac{8192}{15} \]Volume = ∫ A(x) dx across the base.✓ Proved
Answer \( \frac{8192}{15} \)
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 bounds, the side length of the triangular cross-sections, and the area formula for an isosceles right triangle with a leg in the base. The integration is correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the bounds, the side length of the triangular cross-sections, and the area formula for an isosceles right triangle with a leg in the base. The integration is correct.gpt-oss:20b: fail (error) 2026-10-07 — The integral of the cross‑sectional area gives 4096/15, not 8192/15. The solution mistakenly doubles the result. The setup and area formula are correct, but the final evaluation is incorrect.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the bounds of integration, the length of the leg of the triangular cross-section, and the area formula for an isosceles right triangle. The integration is correct.
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 2026-10-07 with SymPy 1.14.0.