Volumes with known cross sections
Problem 5.397 · medium
The base of a solid is the triangle with vertices \( \displaystyle (0,0) \), \( \displaystyle (2,0) \) and \( \displaystyle (0,4) \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are equilateral triangles. Find the volume of the solid.
- \[ 4 - 2 x \]At position x the base of the cross-section has length (top − bottom).✓ Proved
- \[ \frac{\sqrt{3} \left(4 - 2 x\right)^{2}}{4} = \sqrt{3} x^{2} - 4 \sqrt{3} x + 4 \sqrt{3} \]Area of one cross-section: sqrt(3)/4·(side)².✓ Proved
- \[ \int\limits_{0}^{2} \left(\sqrt{3} x^{2} - 4 \sqrt{3} x + 4 \sqrt{3}\right)\, dx = \frac{8 \sqrt{3}}{3} \]Volume = ∫ A(x) dx across the base.✓ Proved
Answer \( \frac{8 \sqrt{3}}{3} \)
✓ 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
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the side length of the cross-sections based on the triangle's geometry, applies the correct area formula for equilateral triangles, and integrates over the correct bounds.
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.