Volumes with known cross sections
Problem 5.256 · medium
The base of a solid is the region between \( \displaystyle y = x \) and \( \displaystyle y = x^2 \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are equilateral triangles. Find the volume of the solid.
- \[ - x^{2} + x = x \left(1 - x\right) \]At position x the base of the cross-section has length (top − bottom).✓ Proved
- \[ \frac{\sqrt{3} x^{2} \left(1 - x\right)^{2}}{4} = \frac{\sqrt{3} x^{4}}{4} - \frac{\sqrt{3} x^{3}}{2} + \frac{\sqrt{3} x^{2}}{4} \]Area of one cross-section: sqrt(3)/4·(side)².✓ Proved
- \[ \int\limits_{0}^{1} \left(\frac{\sqrt{3} x^{4}}{4} - \frac{\sqrt{3} x^{3}}{2} + \frac{\sqrt{3} x^{2}}{4}\right)\, dx = \frac{\sqrt{3}}{120} \]Volume = ∫ A(x) dx across the base.✓ Proved
Answer \( \frac{\sqrt{3}}{120} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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 of integration, the side length of the cross-sections, and the area formula for equilateral triangles. The integration is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the bounds of integration, the side length of the cross-sections, and the area formula for equilateral triangles. The integration is correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the bounds of integration, the side length of the cross-sections, and the area formula for equilateral triangles. The integration is performed correctly.gpt-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.