limit of Riemann sum

consider f(x)=x and partitions P , we want the area under the equation f:

$$
f(x)=x\\ P=\left\{ \frac { b }{ n } ,2\frac { b }{ n } ,3\frac { b }{ n } ,4\frac { b }{ n } ...n\frac { b }{ n }  \right\} \quad a<b,\quad a=0\\ \sum _{ k=1 }^{ n }{ f(x_{ k }).\Delta  } x_{ k }=\sum _{ k=1 }^{ n }{ x_{ k }.\Delta  } x\\ =\sum _{ k=1 }^{ n }{ k } \frac { b }{ n } .\frac { b }{ n } =\sum _{ k=1 }^{ n }{ k } \frac { b^{ 2 } }{ n^{ 2 } } \\ =\frac { b^{ 2 } }{ n^{ 2 } } \sum _{ k=1 }^{ n }{ k } \\ =\frac { b^{ 2 } }{ n^{ 2 } } .\frac { n(n+1) }{ 2 } \\ =\frac { b^{ 2 } }{ n } .\frac { (n+1) }{ 2 } \quad \\ =\frac { b^{ 2 } }{ 2 } (1+\frac { 1 }{ n } )
$$

as norms |P| are equal , then :

$$
\lim _{ n\rightarrow \infty  }{ \frac { b^{ 2 } }{ 2 } (1+\frac { 1 }{ n } ) } \\ =\lim _{ n\rightarrow \infty  }{ \frac { b^{ 2 } }{ 2 } .\lim _{ n\rightarrow \infty  }{ (1+\frac { 1 }{ n } ) }  } \\ =\int _{ 0 }^{ a }{ x.dx } =\frac { b^{ 2 } }{ 2 }
$$

for interval a,b :

$$
\int _{ a }^{ b }{ f(x).dx } =\int _{ a }^{ 0 }{ f(x).dx } +\int _{ 0 }^{ b }{ f(x).dx } \\ =-\int _{ 0 }^{ a }{ f(x).dx } +\int _{ 0 }^{ b }{ f(x).dx } \\ =-\int _{ 0 }^{ a }{ x.dx } +\int _{ 0 }^{ b }{ x.dx } \\ =-\frac { a^{ 2 } }{ 2 } +\frac { b^{ 2 } }{ 2 } \\ \\ \int _{ a }^{ b }{ x.dx } =\frac { b^{ 2 } }{ 2 } -\frac { a^{ 2 } }{ 2 } \quad ,\quad a<b
$$