Definite integrals
Problem 4.111 · medium
Evaluate \( \displaystyle \int_{1}^{e} 4 \sqrt{x} - 2 \, dx \).
- By the Fundamental Theorem of Calculus, Part 2, the integral is F(b) − F(a) for any antiderivative F.Reviewed
- \[ \frac{d}{d x} \left(\frac{8 x^{\frac{3}{2}}}{3} - 2 x\right) = 4 \sqrt{x} - 2 \]F(x) = 8*x**(3/2)/3 - 2*x is an antiderivative: its derivative is the integrand.✓ Proved
- \[ - \left. - 2 x + \frac{8 x^{\frac{3}{2}}}{3} \right|_{\substack{ x=1 }} + \left. \frac{8 x^{\frac{3}{2}}}{3} - 2 x \right|_{\substack{ x=e }} = - 2 e - \frac{2}{3} + \frac{8 e^{\frac{3}{2}}}{3} \]F(b) − F(a).✓ Proved
Answer \( - 2 e - \frac{2}{3} + \frac{8 e^{\frac{3}{2}}}{3} \)
✓ 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.
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: passqwen3.6:27b-mlx: pass — The solution correctly applies the Fundamental Theorem of Calculus. The antiderivative is verified, and the evaluation at the bounds 1 and e is algebraically correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the Fundamental Theorem of Calculus. The antiderivative is verified, and the evaluation at the bounds 1 and e is algebraically correct.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the antiderivative and applies the Fundamental Theorem of Calculus. The algebraic evaluation of the definite integral is correct.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.