Volumes with known cross sections
Problem 5.257 · medium
The base of a solid is the triangle with vertices \( \displaystyle (0,0) \), \( \displaystyle (3,0) \) and \( \displaystyle (0,6) \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are semicircles. Find the volume of the solid.
- \[ 6 - 2 x \]At position x the base of the cross-section has length (top − bottom).✓ Proved
- \[ \frac{\pi \left(6 - 2 x\right)^{2}}{8} = \frac{\pi x^{2}}{2} - 3 \pi x + \frac{9 \pi}{2} \]Area of one cross-section: pi/8·(side)².✓ Proved
- \[ \int\limits_{0}^{3} \left(\frac{\pi x^{2}}{2} - 3 \pi x + \frac{9 \pi}{2}\right)\, dx = \frac{9 \pi}{2} \]Volume = ∫ A(x) dx across the base.✓ Proved
Answer \( \frac{9 \pi}{2} \)
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 length of the base of the semicircular cross-sections as the y-value of the hypotenuse (6 - 2x) and applies the correct area formula for a semicircle (pi/8 * side^2). The integration bounds and result are correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the length of the base of the semicircular cross-sections as the y-value of the hypotenuse (6 - 2x) and applies the correct area formula for a semicircle (pi/8 * side^2). The integration bounds and result are correct.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: inconclusive 2026-10-05 — reviewer returned a non-objectgpt-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.