Accumulation functions
Problem 4.1084 · easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{1}^{x} 3 t^{2} - 2 t \, dt \) as a function of \( \displaystyle x \).
- \[ \frac{d}{d t} \left(t^{3} - t^{2}\right) = 3 t^{2} - 2 t \]An antiderivative of the integrand is t**3 - t**2.✓ Proved
- \[ - \left. t^{3} - t^{2} \right|_{\substack{ t=1 }} + \left. t^{3} - t^{2} \right|_{\substack{ t=x }} = x^{2} \left(x - 1\right) \]F(x) = G(x) − G(a).✓ Proved
- \[ \frac{d}{d x} x^{2} \left(x - 1\right) = 3 x^{2} - 2 x \]F′(x) = f(x), as FTC Part 1 says.✓ Proved
- \[ \left. x^{2} \left(x - 1\right) \right|_{\substack{ x=1 }} = 0 \]And F(1) = 0.✓ Proved
Answer \( F(x) = x^{2} \left(x - 1\right) \)
✓ 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. How we verify
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 from a to three values of x |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the Fundamental Theorem of Calculus to find the antiderivative and evaluate the definite integral. The algebraic simplification is correct, and the verification steps confirm the result.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the Fundamental Theorem of Calculus to find the antiderivative and evaluate the definite integral. The algebraic simplification is correct, and the verification steps confirm the result.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Fundamental Theorem of Calculus Part 2 to evaluate the definite integral as an antiderivative evaluated at the bounds. The algebraic simplification and verification steps are correct.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies an antiderivative, applies the Fundamental Theorem of Calculus to evaluate the definite integral, and verifies the result. The logic is sound and the algebra is correct.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/ftc_accumulation, checked 2026-10-17 with SymPy 1.14.0.