Question

Evaluate: ${\int }_0^2(x^2+3)dx$ as limit of sums

Answer 1

${\int }_0^2(x^2+3)dx$

$=\underset{n\to \infty }{\underset{n\to 0}{lim}}n\left[f\right(0)+f(n)+f(2n)+....+f((n-1)n\left)\right]$

$=\underset{n\to \infty }{\underset{n\to 0}{lim}}n{\sum }_{r=0}^{n-1}f\left(rn\right)$ $n=\frac{2-0}{n}$ $n=\frac{2}{n}$

$=\underset{n\to \infty }{\underset{n\to 0}{lim}}n{\sum }_{r=0}^{n-1}f\left(\frac{2r}{n}\right)$

$=\underset{n\to \infty }{\underset{n\to 0}{lim}}n{\sum }_{r=0}^{n-1}\frac{4r^2}{n^2}+3$

$=\underset{n\to \infty }{\underset{n\to 0}{lim}}\frac{2}{n}[3n+\frac{4}{n^2}\times \frac{(n-1)\times n(2n-1)}{6}]$

$=\underset{n\to \infty }{lim}2[3+4\frac{(1-\frac{1}{n})(2-\frac{1}{x})}{6}]$

$=2(3+\frac{4\times 2}{6})$

$=2(3+\frac{4}{3})$

$2\times \frac{13}{3}=\frac{26}{3}$