Arc length
Problem 5.116 · medium
Find the length of the curve \( \displaystyle y = \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} \) from \( \displaystyle x = 0 \) to \( \displaystyle x = 2 \).
- Arc length is ∫ √(1 + (dy/dx)²) dx.
- \[ \frac{d}{d x} \frac{2 \left(x + 1\right)^{\frac{3}{2}}}{3} = \sqrt{x + 1} \]dy/dx.✓ Proved
- \[ x + 2 \]1 + (dy/dx)² simplifies (here to a perfect square, which is why these are set).✓ Proved
- \[ \int\limits_{0}^{2} \sqrt{x + 2}\, dx = \frac{16}{3} - \frac{4 \sqrt{2}}{3} \]Integrate.✓ Proved
Answer \( \frac{16}{3} - \frac{4 \sqrt{2}}{3} \)
Lines: 3 proved, 1 not checked. 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | 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 |
| 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 | a 200,000-segment polygon along the curve has the same length |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly simplifies 1 + (dy/dx)^2 to x + 2. Since dy/dx = sqrt(x+1), 1 + (dy/dx)^2 = 1 + x + 1 = x + 2 is actually correct algebraically, but the integrand should be sqrt(x+2). Wait, let me re-read. 1. dy/dx = sqrt(x+1). 2. 1 + (dy/dx)^2 = 1 + (sqrt(x+1))^2 = 1 + x + 1 = x + 2. 3. The integrand is sqrt(1 + (dy/dx)^2) = sqrt(x+2). 4. The integral is from 0 to 2 of sqrt(x+2) dx. Let's check the integration. Integral of (x+2)^(1/2) is (2/3)(x+2)^(3/2). Evaluated from 0 to 2: At x=2: (2/3)(4)^(3/2) = (2/3)(8) = 16/3. At x=0: (2/3)(2)^(3/2) = (2/3)(2*sqrt(2)) = 4*sqrt(2)/3. Result: 16/3 - 4*sqrt(2)/3. The solution steps are actually correct. Why did I think it was wrong? Let me re-read the prompt's instruction. "Find what is wrong". Let's look closely at step 3: "x + 2, x + 2 -- 1 + (dy/dx)^2 simplifies". The text says "1 + (dy/dx)^2 simplifies (here to a perfect square, which is why these are set)". Is x+2 a perfect square? No. x+2 is not a perfect square of a polynomial. Usually, these problems are designed so that 1+(y')^2 is a perfect square, e.g., (something)^2, so the square root cancels out nicely. Here, 1+(y')^2 = x+2. The square root is sqrt(x+2). This is not a perfect square in the sense that sqrt(x+2) simplifies to a polynomial. However, the calculation proceeds correctly by integrating sqrt(x+2). Is the statement "simplifies ... to a perfect square" false? Yes. x+2 is not a perfect square. A perfect square would be something like (x+1)^2 or (sqrt(x)+1)^2. The comment implies that the square root operation would remove the square root because the inside is a perfect square. But sqrt(x+2) does not simplify to a rational function or polynomial. This is a misleading statement about the nature of the expression, even if the subsequent integration is handled correctly (by just integrating the square root). Actually, looking at the provided answer: 16/3 - 4*sqrt(2)/3. If the inside were a perfect square, say (x+1)^2, the integral would be of (x+1). Here the integral is of sqrt(x+2). The comment "which is why these are set" is confusing and mathematically inaccurate regarding the term "perfect square". Let's verify if there is a more severe error. Step 2: dy/dx = sqrt(x+1). Correct. Step 3: 1 + (dy/dx)^2 = x+2. Correct. Step 4: Integral sqrt(x+2). Correct. The error is in the descriptive sentence in step 3 claiming x+2 is a perfect square. This is false. x+2 is not a perfect square. This is a "misleading" or "error" in the explanation. Given the strict adversarial role, claiming a non-square is a perfect square is an error in mathematical terminology/fact. Verdict: fail. Severity: misleading (or error). I will choose error because it states a false mathematical fact.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution incorrectly simplifies 1 + (dy/dx)^2 to x + 2. Since dy/dx = sqrt(x+1), 1 + (dy/dx)^2 = 1 + x + 1 = x + 2 is actually correct algebraically, but the integrand should be sqrt(x+2). Wait, let me re-read. 1. dy/dx = sqrt(x+1). 2. 1 + (dy/dx)^2 = 1 + (sqrt(x+1))^2 = 1 + x + 1 = x + 2. 3. The integrand is sqrt(1 + (dy/dx)^2) = sqrt(x+2). 4. The integral is from 0 to 2 of sqrt(x+2) dx. Let's check the integration. Integral of (x+2)^(1/2) is (2/3)(x+2)^(3/2). Evaluated from 0 to 2: At x=2: (2/3)(4)^(3/2) = (2/3)(8) = 16/3. At x=0: (2/3)(2)^(3/2) = (2/3)(2*sqrt(2)) = 4*sqrt(2)/3. Result: 16/3 - 4*sqrt(2)/3. The solution steps are actually correct. Why did I think it was wrong? Let me re-read the prompt's instruction. "Find what is wrong". Let's look closely at step 3: "x + 2, x + 2 -- 1 + (dy/dx)^2 simplifies". The text says "1 + (dy/dx)^2 simplifies (here to a perfect square, which is why these are set)". Is x+2 a perfect square? No. x+2 is not a perfect square of a polynomial. Usually, these problems are designed so that 1+(y')^2 is a perfect square, e.g., (something)^2, so the square root cancels out nicely. Here, 1+(y')^2 = x+2. The square root is sqrt(x+2). This is not a perfect square in the sense that sqrt(x+2) simplifies to a polynomial. However, the calculation proceeds correctly by integrating sqrt(x+2). Is the statement "simplifies ... to a perfect square" false? Yes. x+2 is not a perfect square. A perfect square would be something like (x+1)^2 or (sqrt(x)+1)^2. The comment implies that the square root operation would remove the square root because the inside is a perfect square. But sqrt(x+2) does not simplify to a rational function or polynomial. This is a misleading statement about the nature of the expression, even if the subsequent integration is handled correctly (by just integrating the square root). Actually, looking at the provided answer: 16/3 - 4*sqrt(2)/3. If the inside were a perfect square, say (x+1)^2, the integral would be of (x+1). Here the integral is of sqrt(x+2). The comment "which is why these are set" is confusing and mathematically inaccurate regarding the term "perfect square". Let's verify if there is a more severe error. Step 2: dy/dx = sqrt(x+1). Correct. Step 3: 1 + (dy/dx)^2 = x+2. Correct. Step 4: Integral sqrt(x+2). Correct. The error is in the descriptive sentence in step 3 claiming x+2 is a perfect square. This is false. x+2 is not a perfect square. This is a "misleading" or "error" in the explanation. Given the strict adversarial role, claiming a non-square is a perfect square is an error in mathematical terminology/fact. Verdict: fail. Severity: misleading (or error). I will choose error because it states a false mathematical fact.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly identifies the arc length formula, computes the derivative and the integrand, and evaluates the definite integral with the correct bounds and result.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/arc_length, checked 2026-09-26 with SymPy 1.14.0.