Volumes with known cross sections
Problem 5.255 · medium
The base of a solid is the region under \( \displaystyle y = \sqrt{x} \) for \( \displaystyle 0 \le x \le 16 \). 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
- \[ \int\limits_{0}^{16} x\, dx = 128 \]Volume = ∫ A(x) dx across the base.✓ Proved
Answer \( 128 \)
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 incorrectly identifies the side length of the square cross-section as x instead of sqrt(x). Consequently, it integrates x instead of (sqrt(x))^2 = x, but the logic presented in the sentences is incoherent and does not justify the setup. Specifically, line 1 claims the base length is sqrt(x) but line 2 implies the area is x (which is correct for the area, but the derivation is missing the squaring step explicitly linked to the side length). More critically, the text says 'Area... 1*(side)^2' but the equation is 'x=x', which is a tautology, not a derivation of Area = x. The student needs to see that side = sqrt(x), so Area = (sqrt(x))^2 = x. The current sentences are logically disconnected from the equations.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution incorrectly identifies the side length of the square cross-section as x instead of sqrt(x). Consequently, it integrates x instead of (sqrt(x))^2 = x, but the logic presented in the sentences is incoherent and does not justify the setup. Specifically, line 1 claims the base length is sqrt(x) but line 2 implies the area is x (which is correct for the area, but the derivation is missing the squaring step explicitly linked to the side length). More critically, the text says 'Area... 1*(side)^2' but the equation is 'x=x', which is a tautology, not a derivation of Area = x. The student needs to see that side = sqrt(x), so Area = (sqrt(x))^2 = x. The current sentences are logically disconnected from the equations.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution incorrectly identifies the side length of the square cross-sections as x instead of sqrt(x), leading to the wrong integrand. The area should be (sqrt(x))^2 = x, but the setup implies the side is x, which is dimensionally and geometrically incorrect for the given base.gpt-oss:20b: fail (misleading) 2026-10-05 — The solution’s sentences incorrectly describe the geometry: it states "sqrt(x)=sqrt(x) – At position x the base of the cross‑section has length (top − bottom)" and "x=x – Area of one cross‑section: 1·(side)²" without properly explaining that the side of each square is √x. While the final integral happens to be correct, the misleading wording could confuse a student about how the side length is determined.
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.