Accumulation functions
Problem 4.1087 · easy
Use the Fundamental Theorem of Calculus to write \( \displaystyle F(x) = \displaystyle \int_{0}^{x} t^{3} \, dt \) as a function of \( \displaystyle x \).
- \[ \frac{d}{d t} \frac{t^{4}}{4} = t^{3} \]An antiderivative of the integrand is t**4/4.✓ Proved
- \[ - \left. \frac{t^{4}}{4} \right|_{\substack{ t=0 }} + \left. \frac{t^{4}}{4} \right|_{\substack{ t=x }} = \frac{x^{4}}{4} \]F(x) = G(x) − G(a).✓ Proved
- \[ \frac{d}{d x} \frac{x^{4}}{4} = x^{3} \]F′(x) = f(x), as FTC Part 1 says.✓ Proved
- \[ \left. \frac{x^{4}}{4} \right|_{\substack{ x=0 }} = 0 \]And F(0) = 0.✓ Proved
Answer \( F(x) = \frac{x^{4}}{4} \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt. 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 Part 2 to evaluate the definite integral as a function of x. The steps are logically sound and algebraically correct.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the Fundamental Theorem of Calculus Part 2 to evaluate the definite integral as a function of x. The steps are logically sound and algebraically correct.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the Fundamental Theorem of Calculus to evaluate the definite integral as a function of x. The steps are logically sound and algebraically verified.gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: fail (error) 2026-10-14 — The problem asks to use the Fundamental Theorem of Calculus to write F(x) as a function of x. The solution incorrectly treats the variable of integration 't' as the variable for the antiderivative in step 1 (Derivative(t**4/4, t)), confusing the dummy variable with the limit variable x. While the final answer is correct, the reasoning demonstrates a fundamental misunderstanding of how FTC relates the integrand f(t) to the antiderivative G(x).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.