Double integrals over general regions
Problem 11.78 · easy
Evaluate \( \displaystyle \iint_D 6 x y \, dA \) where D is bounded by \( \displaystyle y = 0 \), \( \displaystyle y = x \), \( \displaystyle x = 0 \) and \( \displaystyle x = 2 \).
- D is Type I: for each x in [0, 2], y runs from 0 to x.
- \[ \int\limits_{0}^{x} 6 x y\, dy = 3 x^{3} \]The inner integral.✓ Proved
- \[ \int\limits_{0}^{2} 3 x^{3}\, dx = 12 \]The outer integral.✓ Proved
Answer \( 12 \)
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 | nested numerical quadrature over the region 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_region, checked 2026-09-26 with SymPy 1.14.0.