# Schwerpunkt/Zwei Mengen/Nullmengenschnitt/Aufgabe/Lösung

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Die ${\displaystyle {}i}$-te Koordinate des Schwerpunktes von ${\displaystyle {}S\cup T}$ ist unter Verwendung von Fakt  (3)

{\displaystyle {}{\begin{aligned}{\frac {\int _{S\cup T}x_{i}d\lambda ^{n}}{\lambda ^{n}(S\cup T)}}&={\frac {\int _{S\cup T}x_{i}d\lambda ^{n}}{\lambda ^{n}(S)+\lambda ^{n}(T)}}\\&={\frac {\int _{S}x_{i}d\lambda ^{n}+\int _{T}x_{i}d\lambda ^{n}}{\lambda ^{n}(S)+\lambda ^{n}(T)}}\\&={\frac {\int _{S}x_{i}d\lambda ^{n}}{\lambda ^{n}(S)+\lambda ^{n}(T)}}+{\frac {\int _{T}x_{i}d\lambda ^{n}}{\lambda ^{n}(S)+\lambda ^{n}(T)}}\\&={\frac {\lambda ^{n}(S)}{\lambda ^{n}(S)+\lambda ^{n}(T)}}\cdot {\frac {\int _{S}x_{i}d\lambda ^{n}}{\lambda ^{n}(S)}}+{\frac {\lambda ^{n}(T)}{\lambda ^{n}(S)+\lambda ^{n}(T)}}\cdot {\frac {\int _{T}x_{i}d\lambda ^{n}}{\lambda ^{n}(T)}}\\&={\frac {\lambda ^{n}(S)}{\lambda ^{n}(S)+\lambda ^{n}(T)}}\cdot P_{i}+{\frac {\lambda ^{n}(T)}{\lambda ^{n}(S)+\lambda ^{n}(T)}}\cdot Q_{i},\end{aligned}}}

wobei ${\displaystyle {}P_{i}}$ (bzw. ${\displaystyle {}Q_{i}}$) die ${\displaystyle {}i}$-te Koordinate von ${\displaystyle {}P}$ (bzw. ${\displaystyle {}Q}$)

bezeichnet.