Volumes with known cross sections
Problem 5.401 · medium
The base of a solid is the triangle with vertices \( \displaystyle (0,0) \), \( \displaystyle (4,0) \) and \( \displaystyle (0,5) \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are isosceles right triangles with a leg in the base. Find the volume of the solid.
- \[ 5 - \frac{5 x}{4} \]At position x the base of the cross-section has length (top − bottom).✓ Proved
- \[ \frac{\left(5 - \frac{5 x}{4}\right)^{2}}{2} = \frac{25 x^{2}}{32} - \frac{25 x}{4} + \frac{25}{2} \]Area of one cross-section: 1/2·(side)².✓ Proved
- \[ \int\limits_{0}^{4} \left(\frac{25 x^{2}}{32} - \frac{25 x}{4} + \frac{25}{2}\right)\, dx = \frac{50}{3} \]Volume = ∫ A(x) dx across the base.✓ Proved
Answer \( \frac{50}{3} \)
✓ 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 — The solution correctly identifies the length of the leg of the cross-sections as the y-value of the hypotenuse of the base triangle, applies the correct area formula for an isosceles right triangle, and integrates over the correct bounds.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the length of the leg of the cross-sections as the y-value of the hypotenuse of the base triangle, applies the correct area formula for an isosceles right triangle, and integrates over the correct bounds.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the length of the leg of the cross-section as the y-value of the hypotenuse line, applies the correct area formula for an isosceles right triangle, 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.