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Definite integrals

Problem 4.59 · medium Video

Evaluate \( \displaystyle \int_{1}^{2} 3 \sqrt{x} - 2 \, dx \).

By the Fundamental Theorem of Calculus, Part 2, the integral is F(b) − F(a) for any antiderivative F.
Reviewed
\[ \frac{d}{d x} \left(2 x^{\frac{3}{2}} - 2 x\right) = 3 \sqrt{x} - 2 \]
F(x) = 2*x**(3/2) - 2*x is an antiderivative: its derivative is the integrand.✓ Proved
\[ - \left. - 2 x + 2 x^{\frac{3}{2}} \right|_{\substack{ x=1 }} + \left. - 2 x + 2 x^{\frac{3}{2}} \right|_{\substack{ x=2 }} = -4 + 4 \sqrt{2} \]
F(b) − F(a).✓ Proved

Answer \( -4 + 4 \sqrt{2} \)

The graph of f(x) = 3sqrt(x) - 2 from x = 1 to x = 2. Rectangles under the curve get thinner until they fill the area, -4 + 4sqrt(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 (mpmath) gives the same value

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the Fundamental Theorem of Calculus, identifies the correct antiderivative, and evaluates the definite integral accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the Fundamental Theorem of Calculus, identifies the correct antiderivative, and evaluates the definite integral accurately.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the Fundamental Theorem of Calculus, identifies the correct antiderivative, and evaluates the definite integral accurately.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the antiderivative and applies the Fundamental Theorem of Calculus. The algebraic evaluation is correct.
gpt-oss:20b: pass 2026-09-26

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/definite_integral, checked 2026-09-28 with SymPy 1.14.0.