Integral/rst+t sin s/Über Einheitswürfel/Aufgabe/Lösung

Aufgrund des Satzes von Fubini ist

{}{\begin{aligned}\int _{W}rst+t\sin s\,d\lambda ^{3}&=\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}{\left(rst+t\sin s\right)}drdsdt\\&=\int _{0}^{1}\int _{0}^{1}{\left({\frac {1}{2}}r^{2}st+rt\sin s\right)}|_{0}^{1}dsdt\\&=\int _{0}^{1}\int _{0}^{1}{\left({\frac {1}{2}}st+t\sin s\right)}dsdt\\&=\int _{0}^{1}{\left({\frac {1}{4}}s^{2}t-t\cos s\right)}|_{0}^{1}dt\\&=\int _{0}^{1}{\left({\frac {1}{4}}t-t\cos 1+t\right)}dt\\&=\int _{0}^{1}{\left({\left({\frac {5}{4}}-\cos 1\right)}t\right)}dt\\&={\left({\frac {1}{2}}{\left({\frac {5}{4}}-\cos 1\right)}t^{2}\right)}|_{0}^{1}\\&={\frac {5}{8}}-{\frac {1}{2}}\cos 1.\end{aligned}}