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Definite integrals from area formulas

Problem 4.1141 · easy Video

Evaluate \( \displaystyle \int_{-1}^{1} \sqrt{1 - x^{2}} \, dx \) using a formula from geometry.

The region between the graph and the x-axis is a semicircle of radius 1.
Reviewed
\[ \frac{1^{2} \pi}{2} = \frac{\pi}{2} \]
Its (signed) area.✓ Proved
\[ \int\limits_{-1}^{1} \sqrt{1 - x^{2}}\, dx = \frac{\pi}{2} \]
Integrating directly gives the same.✓ Proved

Answer \( \frac{\pi}{2} \)

The graph of f(x) = sqrt(1 - x²) from x = -1 to x = 1. Rectangles under the curve get thinner until they fill the area, pi/2.

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
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

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the integrand as the upper semicircle of radius 1 and applies the geometric area formula accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the integrand as the upper semicircle of radius 1 and applies the geometric area formula accurately.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the geometric shape as a semicircle of radius 1 and applies the area formula accurately. The logic is sound and directly answers the prompt.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the geometric interpretation of the integral as the area of a semicircle of radius 1 and applies the correct area formula.
gpt-oss:20b: pass 2026-10-14

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/integral_geometry, checked 2026-10-18 with SymPy 1.14.0.