Volumes with known cross sections
Problem 5.329 · medium
The base of a solid is the region under \( \displaystyle y = \sqrt{x} \) for \( \displaystyle 0 \le x \le 9 \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are squares. Find the volume of the solid.
- \[ \sqrt{x} \]At position x the base of the cross-section has length (top − bottom).✓ Proved
- \[ x \]Area of one cross-section: 1·(side)².✓ Proved
- \[ \int\limits_{0}^{9} x\, dx = \frac{81}{2} \]Volume = ∫ A(x) dx across the base.✓ Proved
Answer \( \frac{81}{2} \)
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: fail (error) — The solution fails to define the area function A(x) correctly. The side length of the square cross-section is sqrt(x), so the area should be (sqrt(x))^2 = x. While the final integral of x from 0 to 9 is correct, the intermediate steps (1 and 2) are tautologies that do not establish the relationship between the geometry and the integrand, making the derivation logically incomplete and misleading.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to define the area function A(x) correctly. The side length of the square cross-section is sqrt(x), so the area should be (sqrt(x))^2 = x. While the final integral of x from 0 to 9 is correct, the intermediate steps (1 and 2) are tautologies that do not establish the relationship between the geometry and the integrand, making the derivation logically incomplete and misleading.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution fails to define the side length of the square cross-sections as y = sqrt(x), resulting in an incorrect integrand of x instead of (sqrt(x))^2 = x. Although the final numerical answer coincidentally matches the correct volume, the derivation is mathematically invalid because it integrates x rather than the area function.
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 2026-10-07 with SymPy 1.14.0.