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

Question

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

✓

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} $

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following:
$\begin{vmatrix}0&\text{xy}^2&\text{xz}^2\\\text{x}^2\text{y}&0&\text{yz}^2\\\text{x}^2\text{z}&\text{zy}^2&0\end{vmatrix}$

Solve the following initial value problems:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}(\text{x}+2\text{y})}{\text{x}(2\text{x}+\text{y})},\text{y}(1)=2$

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,
$\text{f}'(\text{x})=\text{x}^{4}-62\text{x}^{2}+120\text{x}+9$

A manufacturer produces two Models of bikes-Model X and Model Y. Model X takes a 6 man-hours to make per unit, while Model Y takes 10 man-hours per unit. There is a total of 450 man-hour available per week. Handling and Marketing costs are Rs. 2000 and Rs. 1000 per unit for Models X and Y respectively. The total funds available for these purposes are Rs. 80,000 per week. Profits per unit for Models X and Y are Rs. 1000 and Rs. 500, respectively.
How many bikes of each model should the manufacturer produce so as to yield a maximum profit? Find the maximum profit.

Differentiate the following functions with respect to x:
$10^{\log\sin\text{x}}$

Evaluate the following integrals:
$\int\frac{1}{1+\sqrt{\text{x}}}\text{dx}$

Find the second order derivatives of the following functions:
$\text{y}=\tan^{-1}\text{x}$

Find the angle between the pairs of lines with direction ratios proportional to
$2, 2, -2$ and $4, 1, 8$

Maximum Z = 2x + 4y Subject to$\text{x}+\text{y}\geq8$
$\text{x}+4\text{y}\geq12$
$\text{x}\geq3,\text{y}\geq2$

Evaluate the following integrals:
$\int^\limits{\frac{\pi}{2}}_{0}\frac{\tan\text{x}}{1+\text{m}^2\tan^2\text{x}}\text{ dx}$