Evaluate _0^ 2 (x^ 2 +x + a) dx as limit of a sum. - Vidyadip

Question

$\text{Evaluate} \int\limits_0^{2} (x^{2} +x + a) \text{dx as limit of a sum.}$

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Answer

We have to find $\text{I} = \int\limits_0^{2} (x^{2} +x + a) \text{dx as limit of a sum.}$$\text{Here h} = \frac{2 - 0 }{n}= \frac{2}{n}$
$\lim\limits_{X\rightarrow \infty} \text{h} [f(0 +(0 +h) +f(0 +2h) +{\dots}\text{ } { \dots}+ f(0 +\overline{n-1}\text{ h})]$
$= \lim\limits_{X\rightarrow \infty} \frac{2}{n}[1 + (h^{2} +h +1 )+ (4h^{2} + 2h +1) + {\dots\dots}+ (n-1) ^{2}h^{2} + (n-1)h + 1]$
$\lim\limits_{X\rightarrow \infty} \frac{2}{n} \bigg[n +h^{2} \frac{(n-1)n(2n-1)}{6} +\frac{(n-1)n}{2}\bigg]$
$\lim\limits_{X\rightarrow \infty} \frac{2}{n} \bigg[n\frac{4}{6n^{2}}(n-1)(n)(2n-1) +\frac{(n-1)n}{n}\bigg]$
$= 2\bigg[1 + \frac{4}{3} +1\bigg] = \frac{20}{3} $

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