Evaluate the following integrals as limit of sum: ^5_ 0 (x+1)dx - Vidyadip

Question

Evaluate the following integrals as limit of sum:
$\int\limits^5_{0}(\text{x}+1)\text{dx}$

✓

Answer

$\int\limits^{\text{b}}_\text{a}\text{f(x)}\text{dx}=\lim\limits_{\text{h}\rightarrow0}\text{h}\Big[\text{f}(\text{a})+\text{f}(\text{a}+\text{h})+\text{f}(\text{a}+2\text{h})\ +\\ ....\ +\text{f}(\text{a}+(\text{n}-1)\text{h})\Big]$
Where, $\text{h}=\frac{\text{b}-\text{a}}{\text{n}}$
Here, $\text{a}=0,\text{ b}=5,\text{ f(x)}=\text{x}+1,\text{ h}=\frac{5-1}{\text{n}}=\frac{5}{\text{n}}$
Therefore, $\text{I}=\int\limits^5_{0}(\text{x}+1)\text{dx}$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\Big[\text{f}(0)+\text{f}(0+\text{h})+\ ....\ +\text{f}\big\{0+(\text{n}-1)\text{h}\}\Big]$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\Big[(0+1)+(\text{h}+1)+\ ....\ +\{(\text{n}-1)\text{h}+1\}\Big]$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\Big[\text{n}+\text{h}\big\{1+2+3+\ ....\ +(\text{n}-1)\big\}\Big]$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\Big[\text{n}+\text{h}\frac{\text{n}(\text{n}-1)}{2}\Big]$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{5}{\text{n}}\Big[\text{n}+\frac{5\text{n}-5}{2}\Big]$
$=\lim\limits_{\text{n}\rightarrow\infty}5\Big(\frac{7}{2}-\frac{5}{\text{n}}\Big)$
$=\frac{35}{2}$

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 Definite Integrals→Definite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

A manufacturer has three machines installed in his factory. machines $I$ and $II$ are capable of being operated for at most $12$ hours whereas Machine $III$ must operate at least for $5$ hours a day. He produces only two items, each requiring the use of three machines. The number of hours required for producing one unit each of the items on the three machines is given in the following table:

Item	Number of hours required by the machine
	$I$	$II$	$III$
$A$	$1$	$2$	$1$
$B$	$2$	$1$	$\frac{5}{4}$

He makes a profit of $Rs. 6.00$ on item $A$ and $Rs. 4.00$ on item $B$. Assuming that he can sell all that he produces, how many of each item should he produces so as to maximize his profit? Determine his maximum profit. Formulate this $\text{LPP}$ mathematically and then solve it.

$\text{A}=\begin{bmatrix}1&-2&0\\ 2&1&3\\ 0&-2&1\end{bmatrix}\text{and }\text{B}=\begin{bmatrix}7&2&-6\\ -2&1&-3\\ -4&2&5\end{bmatrix}$, find AB. Hence, solve the system of equations:
x - 2y = 10, 2x + y + 3z = 8 and -2y + z = 7

Evaluate the following integrals:
$\int\limits^{\pi}_0\text{x}\log\sin\text{x}\text{ dx}$

Find the vector equation of the plane through the line of intersection of the planes $x + y + z = 1$ and $2x + 3y + 4z = 5$ which is perpendicular to the plane $x - y + z = 0.$

Evaluate the following integrals:
$\int_{0}^\limits{\text{a}}\frac{\text{x}}{\sqrt{\text{a}^2+\text{x}^2}}\text{ dx}$

Let $\text{A} = \text{R} \times \text{R}$ and let$ _*$ be a binary operation on $A$ defined by $\text{(a, b)} {*} \text{(c, d)} = \text{(ad + bc, bd)}$ for all $\text{(a, b), (c, d)} \in \text{R} \times \text{R}.$

Show that $_*$ is commutative on $A.$
Show that $_*$ is associative on $A.$
Find the identity element of $_*$ in $A.$

Evaluate the following intergrals:
$\int\text{e}^\text{ax}\sin(\text{bx}+\text{c})\text{dx}$

Evaluate the following integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\frac{1}{5\cos\text{x}+3\sin\text{x}}\text{ dx}$

Find $\frac{\text{dy}}{\text{dx}}$ in the following cases: $(x + y)^2 = 2axy$

Find the values of a and b so that the function $\text{f(x)}\begin{cases}\text{x}^2+3\text{x}+\text{a}, & \text{if x}\leq1\\\text{bx}+2, & \text{if x} > 1\end{cases}$ is differentiable at each $\text{x}\in\text{R}.$