Download App

Definite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals5 Marks
Question
Evaluate the following integrals as limit of sum:$\int\limits^\text{b}_{\text{a}}\text{e}^{\text{x}}\text{ dx}$
✓

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, $\text{a}=\text{a},\text{ b}=\text{b},\text{ f(x)}=\text{e}^{\text{x}},\text{ h}=\frac{\text{b}-\text{a}}{\text{n}}$
Therefore, $\text{I}=\int\limits^\text{b}_{\text{a}}\text{e}^{\text{x}}\text{ dx}$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\big[\text{f}(\text{a})+\text{f}(\text{a}+\text{h})+\ ....\ +\text{f}\big\{\text{a}+(\text{n}-1)\text{h}\big\}\big]$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\Big[\text{e}^\text{a}+\text{e}^{\text{a}+\text{h}}+\ .....\ +\text{e}^{\{\text{a}+(\text{n}-1)\text{h}\}}\Big]$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\bigg[\text{e}^{\text{a}}\bigg\{\frac{(\text{e}^{\text{h}})^{\text{n}}-1}{\text{e}^{\text{h}}-1}\bigg\}\bigg]$
$=\lim\limits_{\text{h}\rightarrow0}\Big[\text{e}^\text{a}\frac{\text{e}^{\text{b}-\text{a}}-1}{\text{e}^{\text{h}}-1}\Big]$
$=\lim\limits_{\text{h}\rightarrow0}=\Bigg[\frac{\text{e}^\text{b}-\text{e}^{\text{a}}}{\frac{\text{e}^\text{h}-1}{\text{h}}}\Bigg]$
$=\frac{\text{e}^\text{b}-\text{e}^\text{a}}{1}$
$=\text{e}^{\text{b}}-\text{e}^\text{a}$

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

In a simple circult of resistance R, self inductance L and voltage E, the current i at any times t is given by $\text{L}\frac{\text{di}}{\text{dt}}+\text{R}\text{i}=\text{E}.$ If E is constant and initially no current throught the circuit, prove that $\text{i}=\frac{\text{E}}{\text{R}}\left\{1-\text{e}^-(\frac{\text{R}}{\text{L}})\text{t}\right\}.$
→
If $\text{A}=\begin{bmatrix}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix},$ show that $A^{-1} = A^3.$
→
$\int(\text{x}+2)\sqrt{3\text{x}+5}\text{ dx}$
→
A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:
 
 
Product A
Product B
Weekly capacity
Department 1
3
2
130
Department 2
4
6
260
Selling price per unit
Rs. 25
Rs. 30
 
Labour cost per unit
Rs. 16
Rs. 20
 
Raw material cost per unit
Rs. 4
Rs. 4
 
The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a LPP.
→
Solve the following system of equations by matrix method.
$8x + 4y + 3z = 18$
$2x + y + z = 5$
$x + 2y + z = 5$
→
Evaluate the following integrals:$\int\text{x}\tan^2\text{x dx}$
→
Find the area of the region bounded by the parabola $y^2 = 2x + 1$ and the line $x - y - 1 = 0.$
→
Show that $\text{f}(\text{x})=\tan^{-1}(\sin\text{x}+\cos\text{x})$ is a decreasing function on the interval $\Big(\frac{\pi}{4},\frac{\pi}{2}\Big).$
→
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$f(x) = x^2 - 2x + 4$ on $[1, 5]$
→
Evaluate the following intregals:
$\int\frac{1}{1+3\sin^2\text{x}}\text{ dx}$
→