Surface area of revolution
Problem 5.454 · medium
Find the area of the surface obtained by rotating \( \displaystyle y = \sqrt{x} \), \( \displaystyle 1 \le x \le 6 \), about the \( \displaystyle x \)-axis.
- \[ \frac{d}{d x} \sqrt{x} = \frac{1}{2 \sqrt{x}} \]y′.✓ Proved
- Rotating about the x-axis, the radius is y, so S = ∫ 2π·(radius)·√(1 + (y′)²) dx.
- \[ 2 \pi \sqrt{x} \sqrt{1 + \frac{1}{4 x}} = \pi \sqrt{x} \sqrt{4 + \frac{1}{x}} \]Simplify the integrand.✓ Proved
- \[ \int\limits_{1}^{6} \pi \sqrt{x} \sqrt{4 + \frac{1}{x}}\, dx = \frac{5 \pi \left(25 - \sqrt{5}\right)}{6} \]Integrate.✓ Proved
Answer \( \frac{5 \pi \left(25 - \sqrt{5}\right)}{6} \approx 59.5958 \)
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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 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 of the unsimplified integrand |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The simplification in step 3 is algebraically incorrect. The expression 2*pi*sqrt(x)*sqrt(1 + 1/(4*x)) simplifies to pi*sqrt(4x + 1), not pi*sqrt(x)*sqrt(4 + 1/x). While the latter is technically equal to pi*sqrt(4x+1)/sqrt(x) * sqrt(x) = pi*sqrt(4x+1) only if one carefully handles the sqrt(x) term, the written form pi*sqrt(x)*sqrt(4 + 1/x) is misleading and likely leads to integration errors or confusion, but more importantly, the standard simplification is pi*sqrt(4x+1). Let's check the integral of pi*sqrt(4x+1) vs the stated answer. Integral of sqrt(4x+1) is (1/6)(4x+1)^(3/2). Evaluated from 1 to 6: (1/6)(25^(3/2) - 5^(3/2)) = (1/6)(125 - 5sqrt(5)). Multiplied by pi: pi/6(125 - 5sqrt(5)). The stated answer is 5pi/6(25 - sqrt(5)) = pi/6(125 - 5sqrt(5)). So the final answer is actually correct. However, step 3 claims 2*pi*sqrt(x)*sqrt(1 + 1/(4*x)) = pi*sqrt(x)*sqrt(4 + 1/x). Let's verify this equality. LHS = 2pi sqrt(x) sqrt((4x+1)/(4x)) = 2pi sqrt(x) (sqrt(4x+1))/(2sqrt(x)) = pi sqrt(4x+1). RHS = pi sqrt(x) sqrt((4x+1)/x) = pi sqrt(x) (sqrt(4x+1))/sqrt(x) = pi sqrt(4x+1). So the equation in step 3 is actually algebraically correct. Why did I think it was wrong? Ah, sqrt(4 + 1/x) = sqrt((4x+1)/x). So pi*sqrt(x)*sqrt((4x+1)/x) = pi*sqrt(4x+1). The step is correct. The integral of pi*sqrt(4x+1) is indeed the one yielding the correct answer. So the solution is sound.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The simplification in step 3 is algebraically incorrect. The expression 2*pi*sqrt(x)*sqrt(1 + 1/(4*x)) simplifies to pi*sqrt(4x + 1), not pi*sqrt(x)*sqrt(4 + 1/x). While the latter is technically equal to pi*sqrt(4x+1)/sqrt(x) * sqrt(x) = pi*sqrt(4x+1) only if one carefully handles the sqrt(x) term, the written form pi*sqrt(x)*sqrt(4 + 1/x) is misleading and likely leads to integration errors or confusion, but more importantly, the standard simplification is pi*sqrt(4x+1). Let's check the integral of pi*sqrt(4x+1) vs the stated answer. Integral of sqrt(4x+1) is (1/6)(4x+1)^(3/2). Evaluated from 1 to 6: (1/6)(25^(3/2) - 5^(3/2)) = (1/6)(125 - 5sqrt(5)). Multiplied by pi: pi/6(125 - 5sqrt(5)). The stated answer is 5pi/6(25 - sqrt(5)) = pi/6(125 - 5sqrt(5)). So the final answer is actually correct. However, step 3 claims 2*pi*sqrt(x)*sqrt(1 + 1/(4*x)) = pi*sqrt(x)*sqrt(4 + 1/x). Let's verify this equality. LHS = 2pi sqrt(x) sqrt((4x+1)/(4x)) = 2pi sqrt(x) (sqrt(4x+1))/(2sqrt(x)) = pi sqrt(4x+1). RHS = pi sqrt(x) sqrt((4x+1)/x) = pi sqrt(x) (sqrt(4x+1))/sqrt(x) = pi sqrt(4x+1). So the equation in step 3 is actually algebraically correct. Why did I think it was wrong? Ah, sqrt(4 + 1/x) = sqrt((4x+1)/x). So pi*sqrt(x)*sqrt((4x+1)/x) = pi*sqrt(4x+1). The step is correct. The integral of pi*sqrt(4x+1) is indeed the one yielding the correct answer. So the solution is sound.qwen3.6:27b-mlx: fail (error) 2026-10-10 — The simplification in step 3 is algebraically incorrect. The expression 2*pi*sqrt(x)*sqrt(1 + 1/(4*x)) simplifies to pi*sqrt(4x + 1), not pi*sqrt(x)*sqrt(4 + 1/x). Although the final numerical answer happens to be correct for the intended integral, the intermediate step shown is false and would mislead a student about how to simplify the integrand.gpt-oss:20b: pass 2026-10-10
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/surface_area_revolution, checked 2026-10-10 with SymPy 1.14.0.