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 "Solve the Following Question.(5 Marks)" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluate the following integrals as limit of sum:$\int\limits^{\text{b}}_{\text{a}}\text{x}\text{ dx}$

Find a point on the parabola $y = (x − 3)^2$, where the tangent is parallel to the chord joining $(3, 0)$ and $(4, 1)$.

Evaluate the following integrals:$\int\limits^0_{-5}\text{f(x)}\text{dx,}$ Where $\text{f(x)}=|\text{x}|+|\text{x}+2|+|\text{x}+5|$

Find the equation of the tangent line to the curve $y = x^2 - 2x + 7$ which is perpendicular to the line $5y - 15x = 13.$

Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution.

Evaluate the following integrals:$\int\limits^{\frac{\pi}{3}}_{\frac{\pi}{6}}\frac{\sqrt{\tan\text{x}}}{\sqrt{\tan\text{x}}+\sqrt{\cot\text{x}}}\text{ dx}$

Evaluate : $\int_0^4\left(x-x^2\right) \cdot d x$

A firm manufactures two types of products A and B and sells them at a profit of Rs. 5 per unit of type A and Rs 3 per unit of type B. Each product is processed on two machines $M_1$ and $M_2$.One unit of type $A$ requires one minute of processing time on $M _1$ and two minutes of processing time on $M _2$, whereas one unit of type $B$ requires one minute of processing time on $M_1$ and one minute on $M_2$. Machines $M_1$ and $M_2$ are respectively available for at most 5 hours and 6 hours in a day. Find out how many units of each type of product should the firm produce a day in order to maximize the profit. Solve the problem graphically.

Find the equation of the tangent to the curve $x^2+3 y-3=0$, which is parallel to the line $y=4 x-5$.

$\int \sqrt{\frac{9-x}{x}} d x$