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Definite integrals

Problem 4.8 · easy Video

Evaluate \( \displaystyle \int_{0}^{3} - 3 x^{3} - 4 x^{2} - 3 x + 1 \, dx \).

By the Fundamental Theorem of Calculus, Part 2, the integral is F(b) − F(a) for any antiderivative F.
Reviewed
\[ \frac{d}{d x} \left(- \frac{3 x^{4}}{4} - \frac{4 x^{3}}{3} - \frac{3 x^{2}}{2} + x\right) = - 3 x^{3} - 4 x^{2} - 3 x + 1 \]
F(x) = -3*x**4/4 - 4*x**3/3 - 3*x**2/2 + x is an antiderivative: its derivative is the integrand.✓ Proved
\[ - \left. - \frac{3 x^{4}}{4} - \frac{4 x^{3}}{3} - \frac{3 x^{2}}{2} + x \right|_{\substack{ x=0 }} + \left. - \frac{3 x^{4}}{4} - \frac{4 x^{3}}{3} - \frac{3 x^{2}}{2} + x \right|_{\substack{ x=3 }} = - \frac{429}{4} \]
F(b) − F(a).✓ Proved

Answer \( - \frac{429}{4} \)

The graph of f(x) = -3x³ - 4x² - 3x + 1 from x = 0 to x = 3. Rectangles under the curve get thinner until they fill the area, -429/4. Area below the x-axis counts as negative.

✓ 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 computes 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 computes 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 computes the definite integral accurately.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the Fundamental Theorem of Calculus. The antiderivative is verified, and the evaluation at the bounds 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.