Question

Evaluate each of the following integrals by expressing them as a limit of a sum.
${\int }_0^1(2x+1)dx$

Answer 1

${\int }_0^1(2x+1)dx$

Evaluating integral as a limit of sum

${\int }_a^{b}f\left(x\right)d=(b-a){lim}_{n⟶\infty }\frac{1}{n}[f\left(a\right)+f(a+h)+...f(a+(n-1)h)]$

Where $h=\frac{b-a}{n}$

here, $a=0,b=1h=\frac{1}{n}$

$={lim}_{n⟶\infty }\frac{1}{n}[f\left(0\right)+f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+....f\left(\frac{n-1}{n}\right)]={lim}_{n⟶\infty }\frac{1}{n}[1+\frac{2}{n}+1+2.\frac{2}{n}+1+...\frac{2(2-1)}{n}+1]={lim}_{n⟶\infty }\frac{1}{n}[(1+1+...1)+\frac{2}{n}(1+2+...n-1)]={lim}_{n⟶\infty }\frac{1}{n}[n+\frac{2}{n}\times \frac{(n-1)\left(n\right)}{2}]={lim}_{n⟶\infty }\frac{1}{n}(2n-1)={lim}_{n⟶\infty }\frac{1}{n}(2-\frac{1}{n})=2$

Answer