Home›Calculus 2›Centers of mass and centroids›Problem 5.211

Centers of mass and centroids

Problem 5.211 · medium

Find the centroid of the region bounded by \( \displaystyle y = \sqrt{x} \) and \( \displaystyle y = 0 \) for \( \displaystyle 0 \le x \le 4 \).

\[ \int\limits_{0}^{4} \sqrt{x}\, dx = \frac{16}{3} \]
The area.✓ Proved
\[ \int\limits_{0}^{4} x^{\frac{3}{2}}\, dx = \frac{64}{5} \]
The moment about the y-axis.✓ Proved
\[ \int\limits_{0}^{4} \frac{x}{2}\, dx = 4 \]
The moment about the x-axis: ∫ ½(top² − bottom²) dx.✓ Proved
\[ \left[\begin{matrix}\frac{12}{5}\\\frac{3}{4}\end{matrix}\right] \]
Divide each moment by the area.✓ Proved

Answer \( (\bar{x}, \bar{y}) = \left(\frac{12}{5}, \frac{3}{4}\right) \)

✓ 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
4	✓ 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 each moment

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the area and moments, and the final division yields the correct centroid coordinates. The setup and calculations are sound.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the area and moments, and the final division yields the correct centroid coordinates. The setup and calculations are sound.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04
gpt-oss:20b: pass 2026-10-04

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/center_of_mass, checked 2026-10-04 with SymPy 1.14.0.