Volumes with known cross sections
Problem 5.539 · medium
The base of a solid is the triangle with vertices \( \displaystyle (0,0) \), \( \displaystyle (1,0) \) and \( \displaystyle (0,5) \). Cross-sections perpendicular to the \( \displaystyle x \)-axis are squares. Find the volume of the solid.
- \[ 5 - 5 x \]At position x the base of the cross-section has length (top − bottom).✓ Proved
- \[ \left(5 - 5 x\right)^{2} = 25 x^{2} - 50 x + 25 \]Area of one cross-section: 1·(side)².✓ Proved
- \[ \int\limits_{0}^{1} \left(25 x^{2} - 50 x + 25\right)\, dx = \frac{25}{3} \]Volume = ∫ A(x) dx across the base.✓ Proved
Answer \( \frac{25}{3} \)
✓ 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 — The solution correctly identifies the side length of the square cross-sections as the height of the triangle at position x, sets up the integral for volume correctly, and computes the result accurately.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-13qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the side length of the square cross-sections as the height of the triangle at position x, sets up the integral for volume correctly, and computes the result accurately.qwen3.6:27b-mlx: pass 2026-10-13 — The solution correctly identifies the side length of the square cross-sections as the height of the triangle at position x, sets up the integral of the area function over the correct bounds, and computes the volume accurately.gpt-oss:20b: pass 2026-10-13
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.