Question

Помогите решить двойной интеграл!!!!!!!(очень срочно)

Первое задание:∫∫(x^2y^2+y^2)dxdy ,область D:{1/x≤y≤2/x;x≤y≤3x}

Answer 1

Точки пересечения:

$D:; ; left { {{frac{1}{x}leq yleq 2} atop {xleq yleq 3x}} right.\\left { {{y=3x} atop {y=1/x}} right.; ; to ; ; 3x=frac{1}{x}; ,; ; frac{3x^2-1}{x}=0; ,; ; 3x^2=1; ,; x=pm frac{1}{sqrt3}\\left { {{y=3x} atop {y=2/x}} right.; ; to ; ; 3x=frac{2}{x}; ,; ; frac{3x^2-2}{x}=0; ,; ; 3x^2=2; ,; ; x=pm sqrt{frac{2}{3}}\\left { {{y=x} atop {y=1/x}} right.; ; to ; ; x=frac{1}{x}; ,; ; frac{x^2-1}{x}=0; ,; ; x^2=1; ,; ; x=pm 1$

$left { {{y=x} atop {y=2/x}} right. ; ,; ; x=frac{2}{x}; ,; ; frac{x^2-2}{x}=0; ,; ; x^2=2; ,; ; x=pm sqrt2\\\iint limits _{D}, (x^2y^2+y^2)dx, dy=intlimits_{1/sqrt3}^{sqrt{2/3}}, dxint limits _{1/x}^{3x}(x^2y^2+y^2), dy+\\+intlimits_{sqrt{2/3}}^1, dxint limits _{1/x}^{2/x}, (x^2y^2+y^2), dy+intlimits^{sqrt2}_1, dxint limits _{x}^{2/x}, (x^2y^2+y^2)dx, dy=\\=intlimits^{sqrt{2/3}}_{1|sqrt3}, ((x^2+1)cdot frac{y^3}{3})Big |_{1|x}^{3x}, dx+intlimits^1_{sqrt{2/3}}, ((x^2+1)cdot frac{y^3}{3})Big |_{1|x}^{2|x}, dx+\\+intlimits_1^{sqrt2}, ((x^2+1)cdot frac{y^3}{3})Big |_{x}^{2|x}, dx=$

$=frac{1}{3}intlimits^{sqrt{2/3}}_{1/sqrt3}, (x^2+1)(27x^3-1), dx+frac{1}{3}intlimits^1_{sqrt{2/3}}, (x^2+1)(frac{8}{x^3}-frac{1}{x^3}), dx+\\+frac{1}{3} intlimits^{sqrt2}_1, (x^2+1)(frac{8}{x^3}-x^3), dx=\\=frac{1}{3}intlimits^{sqrt{2/3}}_{1/sqrt3}, (27x^5-x^2+27x^3-1), dx+frac{1}{3}intlimits^1_{sqrt{2/3}}, (frac{7}{x}+frac{7}{x^3}), dx+\\+frac{1}{3}int limits _{1}^{sqrt2}, (frac{8}{x}-x^5+frac{8}{x^3}-x^3), dx=$

$=frac{1}{3}cdot Big (frac{9x^6}{2}-frac{x^3}{3}+frac{27x^4}{4}-xBig )Big |_{1/sqrt3}^{sqrt{2/3}}+frac{7}{3}cdot Big (ln|x|-frac{1}{2x^2}Big )Big |_{sqrt{2/3}}^1+\\+frac{1}{3}cdot Big (8ln|x|-frac{x^6}{6}-frac{4}{x^2}-frac{x^4}{4}Big )Big |_1^{sqrt2}=\\=frac{1}{3}cdot Big (frac{4}{3}-frac{2sqrt2}{9sqrt3}+3-sqrt{frac{2}{3}}-frac{1}{6}+frac{1}{9sqrt3}-frac{3}{4}+frac{1}{sqrt3}Big )+frac{7}{3}cdot Big (-frac{1}{2}-lnsqrt{frac{2}{3}}+frac{3}{4}Big )+$

$+frac{1}{3}cdot Big (8lnsqrt2-frac{8}{6}-2-1+frac{1}{6}+4+frac{1}{4}Big )=\\=frac{1}{3}cdot Big (frac{37}{3}-frac{11}{9}cdot sqrt{frac{2}{3}}+frac{10}{9sqrt3}Big )+frac{7}{3}cdot Big (frac{1}{4}-lnsqrt{frac{2}{3}}Big )+frac{1}{3}cdot Big (8lnsqrt2+frac{1}{12}Big )=\\=frac{85}{18}+frac{10-11sqrt2}{27sqrt3}-frac{7}{6}cdot lnfrac{2}{3}+frac{8sqrt2}{3}$