Doppelintegral/Oberer Halbkreis/x^2y+xy^3/Beispiel

Es sei $T$ die obere Einheitskreishälfte und

$f:T⟶ℝ,(x,y)⟼f(x,y)=x^{2}y+x{y}^3.$

Dann ist nach Fakt

$\begin{array}{cc}{\int }_{T}fd{\lambda }^2& ={\int }_{-1}^1({\int }_0^{\sqrt{1-x^2}}{x}^{2}y+x{y}^{3}dy)dx\\ & ={\int }_{-1}^1(\frac{1}{2}{x}^2{y}^2+\frac{1}{4}x{y}^4){|}_0^{\sqrt{1-x^2}}dx\\ & ={\int }_{-1}^1(\frac{1}{2}{x}^2{\sqrt{1-x^2}}^2+\frac{1}{4}x{\sqrt{1-x^2}}^4)dx\\ & ={\int }_{-1}^1(\frac{1}{2}{x}^2(1-x^2)+\frac{1}{4}x{(1-x^2)}^2)dx\\ & ={\int }_{-1}^1(\frac{1}{2}{x}^2-\frac{1}{2}{x}^4+\frac{1}{4}x-\frac{1}{2}{x}^3+\frac{1}{4}{x}^5)dx\\ & =(\frac{1}{8}{x}^2+\frac{1}{6}{x}^3-\frac{1}{8}{x}^4-\frac{1}{10}{x}^5+\frac{1}{24}{x}^6){|}_{-1}^1\\ & =\frac{2}{15}.\end{array}$