Evaluate _0^ 2 (x^ 2 + 2x + 1) dx as limit of sums. - Vidyadip

Question

$\text{Evaluate} \int\limits_0^{2} (x^{2} + 2x + 1) \text{dx as limit of sums.}$

✓

Answer

$\text{Here h} = \frac{2 - 0}{n} =\frac{2}{n} , \text{f (x)} = x^{2} + 2x + 1$$\therefore \text{I} = \lim\limits_{ h\rightarrow 0} \bigg[ \text{f (0)} + \text{f (h)} + \text{f (2h)} +\dots\dots\dots\dots\text{+} \text{f} \overline{(\text{n} - 1)}\text{h}\bigg]$
$= \lim\limits_{ h\rightarrow 0} \text{h} \bigg[1 + (h^{2} + 2h + 1) + (2^{2}h^{2} + 2.2h + 1) + \dots\dots\dots(n- 1^{2}) h^{2} + 2 (n - 1) h + 1)\bigg]$
$= \lim\limits^{n \rightarrow \infty}_{h \rightarrow 0} \frac{2}{n} \Big[(1+1+1+1+1+\ ........)+\text{h}^2\big(1+2^2+3^2+\ ....\big)\Big]$
$= \lim\limits_{n \rightarrow \infty} \frac{2}{n} \bigg[n +\frac{4}{n^{2}} \frac{(n - 1)n(2n -1)}{6} + \frac{4}{n} .\frac{(n - 1)n}{n}\bigg]$
$= \lim\limits_{n \rightarrow \infty} \Big[2 + \frac{8}{6} \frac{n - 1}{n} \frac{2n - 1}{n} + 4\frac{n - 1}{n}\Big]$
$= 2 + 4 +\frac{8}{3} = \frac{26}{3}$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

(Pythagoras's theorem) Prove by vector method that in a right angleg triang, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Find the area of the region bounded by the ellipse $\frac{{{x^2}}}{4} + \frac{{{y^2}}}{9} = 1$.

Evaluate the following integrals:
$\int\frac{1}{(\text{x}^2+2\text{x}+10)^2}\text{ dx}$

Solve the following differential equation:
$\text{x}^2\frac{\text{dy}}{\text{dx}}=\text{x}^2-2\text{y}^2+\text{xy}$

If either $\vec{\text{a}}=\vec{0}$ or $\vec{\text{b}}=\vec{0},$ then $\vec{\text{a}}.\vec{\text{b}}=0.$ But the converse need not be true. Justify your answer with an example.

Solve the following differential equation $(\text{x}^3+\text{x}^2+\text{x}+1)\frac{\text{dy}}{\text{dx}}=2\text{x}^2+\text{x}$

Find one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)$\text{e}^{-\text{y}}\sec^2\text{y dy}=\text{dx}+\text{x dy}$

A firm manufactures two types of products $A$ and $B$ and sells them at a profit of $Rs.2$ on type $A$ and $Rs.3$ on type $B.$ Each product is processed on two machines $M_1$ and $M_2.$ Type $A$ requires one minute of processing time on $M_1$ and two minutes of $M_2;$ type $B$ requires one minute on $M_1$ and one minute on $M_2.$ The machine $M_1$ is available for not more than $6$ hours $40$ minutes while machine $M_2$ is available for 10 hours during any working day. Formulate the problem as a $LPP.$

Write a value of $\int\text{e}^{\log\sin\text{x}}\cos\text{x}\text{ dx}$

If $\text{y}=\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big)+\sec^{-1}\Big(\frac{1+\text{x}^2}{1-\text{x}^2}\Big),\text{x}>0,$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{4}{1+\text{x}^2}$