Evaluate the following integrals as limit of sum: ^ 4 _ 0 (x+e^ 2… - Vidyadip

Question

Evaluate the following integrals as limit of sum:
$\int\limits^{4}_{0}\big(\text{x}+\text{e}^{2\text{x}}\big)\text{dx}$

✓

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, $\text{a}=0,\text{ b}=4,\text{ f(x)}=\text{x}+\text{e}^{2\text{x}},\text{ h}=\frac{4-0}{\text{n}}=\frac{4}{\text{n}}$
Therefore, $\text{I}=\int\limits^{4}_{0}\big(\text{x}+\text{e}^{2\text{x}}\big)\text{dx}$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\Big[\text{f}(0)+\text{f}(0+\text{h})+\ ....\ +\text{f}\big\{0+(\text{n}-1)\text{h}\}\Big]$
$=\lim\limits_{\text{h}\rightarrow0}\Big[(0+\text{e}^0)+(\text{h}+\text{e}^{2\text{h}})+\ .....\ +\Big\{(\text{n}-1)\text{h}+\text{e}^{2(\text{n}-1)\text{h}}\Big\}\Big]$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\Big[\text{h}\big\{1+2+\ ...+ (\text{n}-1)\text{h}\big\}+\text{e}^0+\text{e}^{2\text{h}}+\text{e}^{4\text{h}}+\ ....+\ \text{e}^{2(\text{n}-1)\text{h}}\Big]$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\bigg[\text{h}\frac{\text{n}(\text{n}-1)}{2}+\frac{(\text{e}^{2\text{h}})^\text{n}-1}{\text{e}^{2\text{h}}-1}\bigg]$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{16}{\text{n}^2}\times\frac{\text{n}(\text{n}-1)}{2}+\lim\limits_{\text{h}\rightarrow0}\frac{\text{e}^8-1}{\frac{\text{e}^{2\text{h}}-1}{\text{h}}}$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{16}{\text{n}^2}\times\frac{\text{n}(\text{n}-1)}{2}+\lim\limits_{\text{h}\rightarrow0}\frac{\text{e}^8-1}{\frac{2(\text{e}^{2\text{h}}-1)}{\text{h}}}$
$=8+\frac{\text{e}^8-1}{2}$
$=\frac{15+\text{e}^8}{2}$

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

Similar questions

If x = a ( θ – sin θ ), y = a (1 + cos θ ), find $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}$.

show that the semi-vertical angle of the cone of the maximum volume and of given slant height $\cos^{-1}\frac{1}{\sqrt{3}}.$

The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a trouser is 0.3, and the probability that he will buy a shirt given that he buys a trouser is 0.4. Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.

A firm manufactures two products A and B. Each product is processed on two machines $M_1$ and $M_2$. Product A requires 4 minutes of processing time on $M_1$ and 8 min. on $M_2$; product B requires 4 minutes on $M_1$ and 4 min. on $M_2$. The machine $M_1$ is available for not more than 8 hrs 20 min. while machine $M_2$ is available for 10 hrs. during any working day. The products A and B are sold at a profit of Rs. 3 and Rs. 4 respectively.
Formulate the problem as a linear programming problem and find how many products of each type should be produced by the firm each day in order to get maximum profit.

Find the area of the region $\left\{(x, y): 0 \leqslant y \leqslant\left(x^2+1\right), 0 \leqslant y \leqslant(x+1), 0 \leqslant x \leqslant 2\right\}$

Find the equation of tangents to the curve $y = x^{3} + 2x - 4,$ which are perpendicular to line $x + 14y + 3 = 0.$

A firm manufacturing two types of electric items, A and B, can make a profit of Rs. 20 per unit of A and Rs. 30 per unit of B. Each unit of A requires 3 motors and 4 transformers and each unit of B requires 2 motors and 4 transformers. The total supply of these per month is restricted to 210 motors and 300 transformers. Type B is an export model requiring a voltage stabilizer which has a supply restricted to 65 units per month. Formulate the linear programing problem for maximum profit and solve it graphically.

Find the area of the smaller region bounded by the ellipse $\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1$ and the line $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=1.$

Using direction ratios show that the points A(2, 3, -4) B(1, - 2, 3) and C(3, 8, -11) are collinear.

Evaluate the following integrals:
$\int\frac{\text{x}^2-1}{\text{x}^4+1}\ \text{dx}$