Quader/Monom/Integration/Aufgabe/Lösung

Die Funktion ${\displaystyle {}f}$ ist ein Produkt von ${\displaystyle {}n}$ Funktionen, nämlich ${\displaystyle {}x_{i}^{d_{i}}}$, die jeweils nur von der einen Variablen ${\displaystyle {}x_{i}}$ abhängen. Daher kann man (eine geeignete Verallgemeinerung von) Fakt anwenden und erhält

${\displaystyle {}{\begin{aligned}\int _{Q}fd\lambda ^{n}&={\left(\int _{a_{1}}^{b_{1}}x_{1}^{d_{1}}dx_{1}\right)}{\left(\int _{a_{2}}^{b_{2}}x_{2}^{d_{2}}dx_{2}\right)}\cdots {\left(\int _{a_{n}}^{b_{n}}x_{n}^{d_{n}}dx_{n}\right)}\\&={\frac {1}{d_{1}+1}}{\left(b_{1}^{d_{1}+1}-a_{1}^{d_{1}+1}\right)}{\frac {1}{d_{2}+1}}{\left(b_{2}^{d_{2}+1}-a_{2}^{d_{2}+1}\right)}\cdots {\frac {1}{d_{n}+1}}{\left(b_{n}^{d_{n}+1}-a_{n}^{d_{n}+1}\right)}\\&={\frac {1}{(d_{1}+1)(d_{2}+1)\cdots (d_{n}+1)}}{\left(b_{1}^{d_{1}+1}-a_{1}^{d_{1}+1}\right)}{\left(b_{2}^{d_{2}+1}-a_{2}^{d_{2}+1}\right)}\cdots {\left(b_{n}^{d_{n}+1}-a_{n}^{d_{n}+1}\right)}.\end{aligned}}}$