Question

вычислить двойной интеграл ∫∫xy² D: xy=4 , x+y-5=0

Answer 1

$\displaystyle \int\limits^4_1dx\int\limits^{5-x}_{\frac{4}{x}}xy^2dy=\int\limits^4_1\Bigg(\frac{xy^3}{3}\bigg|^{5-x}_{\frac{4}{x}}\Bigg)dx=\int\limits^4_1\Bigg(\frac{x(5-x)^3}{3}-\frac{x\cdot (\frac{4}{x})^3}{3}\Bigg)dx=\\ \\ \\ =\int\limits^4_1\frac{-x^4+15x^3-75x^2+125x}{3}dx-\int\limits^4_1\frac{64}{3x^2}dx=\\ \\ \\ =\Bigg(\frac{64}{3x}-\frac{4x^5-75x^4+500x^3-1250x^2}{60}\Bigg)\Bigg|^4_1=\frac{64}{3\cdot 4}-\\\\ -\frac{4\cdot 4^5-75\cdot 4^4+500\cdot 4^3-1250\cdot 4^2}{60}-\frac{64}{3\cdot 1}+\frac{4-75+500-1250}{60}=\frac{441}{20}$