Volumes with known cross sections
Problem 5.534 · medium
The base of a solid is the triangle with vertices \( \displaystyle (0,0) \), \( \displaystyle (2,0) \) and \( \displaystyle (0,3) \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are isosceles right triangles with a leg in the base. Find the volume of the solid.
- \[ 3 - \frac{3 x}{2} \]At position x the base of the cross-section has length (top − bottom).✓ Proved
- \[ \frac{\left(3 - \frac{3 x}{2}\right)^{2}}{2} = \frac{9 x^{2}}{8} - \frac{9 x}{2} + \frac{9}{2} \]Area of one cross-section: 1/2·(side)².✓ Proved
- \[ \int\limits_{0}^{2} \left(\frac{9 x^{2}}{8} - \frac{9 x}{2} + \frac{9}{2}\right)\, dx = 3 \]Volume = ∫ A(x) dx across the base.✓ Proved
Answer \( 3 \)
✓ 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 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the length of the leg of the cross-section as the y-value of the hypotenuse line connecting (0,3) and (2,0). The area formula for an isosceles right triangle with leg length L is L^2/2, which is applied correctly. The integration bounds and result are 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 2026-10-13 with SymPy 1.14.0.