Double integrals over rectangles
Problem 11.69 · easy
Evaluate \( \displaystyle \iint_R x y \, dA \) where \( \displaystyle R = [0, 3] \times [0, 3] \).
- By Fubini's theorem, integrate in y first, then in x.
- \[ \int\limits_{0}^{3} x y\, dy = \frac{9 x}{2} \]The inner integral.✓ Proved
- \[ \int\limits_{0}^{3} \frac{9 x}{2}\, dx = \frac{81}{4} \]The outer integral.✓ Proved
Answer \( \frac{81}{4} \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
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 | 2-D numerical quadrature (mpmath) agrees |
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_rectangle, checked 2026-09-26 with SymPy 1.14.0.