Volumes with known cross sections
Problem 5.535 · medium
The base of a solid is the region under \( \displaystyle y = \sqrt{x} \) for \( \displaystyle 0 \le x \le 1 \). 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
- \[ \int\limits_{0}^{1} x\, dx = \frac{1}{2} \]Volume = ∫ A(x) dx across the base.✓ Proved
Answer \( \frac{1}{2} \)
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 incorrectly identifies the side length of the square cross-sections as x instead of sqrt(x). Since the base is the region under y = sqrt(x) from y=0, the side length is sqrt(x), making the area A(x) = (sqrt(x))^2 = x. While the final integral setup Integral(x, 0, 1) is coincidentally correct for the area, the reasoning in step 1 and 2 is flawed because it implies the side length is x directly or confuses the function with its square. Specifically, step 1 says 'base ... has length sqrt(x)' but step 2 uses 'x' as the side squared? No, step 2 says 'x = x' labeled as 'Area ... 1*(side)^2'. This is incoherent. If side is sqrt(x), area is x. The equation 'x=x' is a tautology that doesn't explain the area derivation. The comment 'Area of one cross-section: 1*(side)^2' is generic, but the equation 'x=x' does not represent the area calculation A(x) = (sqrt(x))^2 = x clearly. It skips the crucial step of squaring the side length. A student might think the side length is x. Actually, looking closer: Step 1 says side is sqrt(x). Step 2 says x=x. This implies Area = x. But it doesn't show Area = (side)^2. It just asserts x=x. This is a 'style' or 'misleading' issue at best, but technically the logic is missing the squaring step. However, the prompt asks to find what is wrong. The derivation is opaque. Let's look at the 'error' severity. Is it wrong? The final answer is correct. The steps are: 1. Side = sqrt(x). 2. Area = x (implied by x=x? No, x=x is just identity). 3. Integral of x is 1/2. The connection between 1 and 2 is missing. The sentence 'Area of one cross-section: 1*(side)^2' is a general formula, but the equation 'x=x' doesn't apply it. It should be 'x = (sqrt(x))^2'. By writing 'x=x', it fails to demonstrate the application of the area formula to the specific side length. This is misleading because it hides the squaring operation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution incorrectly identifies the side length of the square cross-sections as x instead of sqrt(x). Since the base is the region under y = sqrt(x) from y=0, the side length is sqrt(x), making the area A(x) = (sqrt(x))^2 = x. While the final integral setup Integral(x, 0, 1) is coincidentally correct for the area, the reasoning in step 1 and 2 is flawed because it implies the side length is x directly or confuses the function with its square. Specifically, step 1 says 'base ... has length sqrt(x)' but step 2 uses 'x' as the side squared? No, step 2 says 'x = x' labeled as 'Area ... 1*(side)^2'. This is incoherent. If side is sqrt(x), area is x. The equation 'x=x' is a tautology that doesn't explain the area derivation. The comment 'Area of one cross-section: 1*(side)^2' is generic, but the equation 'x=x' does not represent the area calculation A(x) = (sqrt(x))^2 = x clearly. It skips the crucial step of squaring the side length. A student might think the side length is x. Actually, looking closer: Step 1 says side is sqrt(x). Step 2 says x=x. This implies Area = x. But it doesn't show Area = (side)^2. It just asserts x=x. This is a 'style' or 'misleading' issue at best, but technically the logic is missing the squaring step. However, the prompt asks to find what is wrong. The derivation is opaque. Let's look at the 'error' severity. Is it wrong? The final answer is correct. The steps are: 1. Side = sqrt(x). 2. Area = x (implied by x=x? No, x=x is just identity). 3. Integral of x is 1/2. The connection between 1 and 2 is missing. The sentence 'Area of one cross-section: 1*(side)^2' is a general formula, but the equation 'x=x' doesn't apply it. It should be 'x = (sqrt(x))^2'. By writing 'x=x', it fails to demonstrate the application of the area formula to the specific side length. This is misleading because it hides the squaring operation.gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: fail (error) 2026-10-13 — The solution incorrectly identifies the side length of the square cross-sections as x instead of sqrt(x). Consequently, it integrates x instead of (sqrt(x))^2 = x, leading to the wrong integrand and a logically flawed derivation.
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.