Evaluate _ 0 ^ 2 (x^ 2 +x+2)dx as limit of sums. - Vidyadip

Question

Evaluate $\int\limits_{0}^{2}\text{(x}^{2}+\text{x}+2)\text{dx}$ as limit of sums.

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Answer

$\int\limits_0^2\text{f(x) dx}=\lim_{\substack{ \text{h}\rightarrow0\\ \text{or n}\rightarrow\infty }} \text{h[f (0) + f(0 + h) + f (0 + 2h) + ...........+ f (0 + (n - 1)h]} $
where $f(x) = x^2 + x + 2, \text{h}=\frac{2}{\text{n}}$
$=\lim\limits_{\text{n}\rightarrow\infty}\text{h [ 2 + (h}^{2}\text{h + 2) + (2}^{2}\text{h}^{2}+\text{2h + 2) +......+((n - 1)}^{2}\text{h}^{2}+\text{(n - 1) h + 2)}]$
$=\lim\limits_{\text{n}\rightarrow\infty}\text{h [2n + h}^{2}\text{(1}^{2}+2^{2}+3^{3}+........(\text{n - 1))}^{2}+\text{h}(1+2+3+......+\text{(n - 1))}]$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{2}{\text{n}}\Bigg[\text{2n}+\frac{4}{\text{n}^{2}}\cdot\frac{\text{n(n - 1)(2n - 1)}}{6}+\frac{2}{\text{n}}\cdot\frac{\text{n (n -1)}}{2}\Bigg]$
$=\lim\limits_{\text{n}\rightarrow\infty}2\Bigg[2+\frac{2}{3}\Bigg(1-\frac{1}{\text{n}}\Bigg)\Bigg(2-\frac{1}{\text{n}}\Bigg)+\Bigg(1-\frac{1}{\text{n}}\Bigg)\Bigg]$
$=2\Bigg[2+\frac{4}{3}+1\Bigg]=\frac{26}{3}.$

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