Double integrals in polar coordinates
Problem 11.31 · medium
Evaluate \( \displaystyle \iint_D e^{- x^{2} - y^{2}} \, dA \) over the disk \( \displaystyle x^2 + y^2 \le 1 \) using polar coordinates.
- In polar coordinates the disk is 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π, and dA = r dr dθ.
- \[ e^{- r^{2} \sin^{2}{\left(\theta \right)} - r^{2} \cos^{2}{\left(\theta \right)}} = e^{- r^{2}} \]The integrand in r and θ.✓ Proved
- \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{1} r e^{- r^{2}}\, dr\, d\theta = - \frac{\pi}{e} + \pi \]Integrate, remembering the extra r.✓ Proved
Answer \( - \frac{\pi}{e} + \pi \)
Lines: 2 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. How we verify
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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the integral done in x and y, numerically, without polar coordinates, agrees |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution incorrectly states the radial bound as R instead of 1 in the first sentence, although the subsequent integration correctly uses 1. This inconsistency between the setup description and the calculation is an error.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution incorrectly states the radial bound as R instead of 1 in the first sentence, although the subsequent integration correctly uses 1. This inconsistency between the setup description and the calculation is an error.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the polar coordinate transformation, bounds, and Jacobian. The integration steps are algebraically correct and lead to the right answer.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly sets up the polar coordinates, transforms the integrand, and evaluates the integral.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/double_integral_polar, checked 2026-09-28 with SymPy 1.14.0.