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:$\int\limits^{2}_{0}\big(\text{x}^2+\text{x}\big)\text{dx}$
✓

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, $\text{a}=0,\text{ b}=2,\text{ f(x)}=\text{x}^2+\text{x},\text{ h}=\frac{2-0}{\text{n}}=\frac{2}{\text{n}}$
Therefore, $\text{I}=\int\limits^{2}_{0}\big(\text{x}^2+\text{x}\big)\text{dx}$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\Big[\text{f}(0)+\text{f}(0+\text{h})+\ ....\ +\text{f}\big\{0+(\text{n}-1)\text{h}\}\Big]$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\big[(0+0)+(\text{h}^2+\text{h})+\ ....\ +\big\{(\text{n}-1)^2\text{h}^2+\text{h}\big\}\big]$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\Big[\text{h}^2\big(1^2+2^2+3^2\ ....\ +(\text{n}-1)^2\big)+\\\text{h}\big\{1+2+3+\ ...\ +(\text{n}-1)\text{h}\big\}\Big]$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\bigg[\text{h}^2\frac{\text{n}(\text{n}-1)(2\text{n}-1)}{6}+\text{h}\frac{\text{n}(\text{n}-1)}{2}\bigg]$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{2}{\text{n}}\Big[\frac{2\text{n}(\text{n}-1)(2\text{n}-1)}{3\text{n}}+\text{n}-1\Big]$
$=\lim\limits_{\text{n}\rightarrow\infty}2\Big[\frac{2}{3}\Big(1-\frac{1}{\text{n}}\Big)\Big(2-\frac{1}{\text{n}}\Big)+1-\frac{1}{\text{n}}\Big]$
$=\frac{8}{3}+2$
$=\frac{14}{3}$

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 "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

ABCD are four points in a plane and Q is the point of intersection of the lines joining the mid-points of AB and CD, BC and AD. Show that $\overrightarrow{\text{PA}}+\overrightarrow{\text{PB}}+\overrightarrow{\text{PC}}+\overrightarrow{\text{PD}}=4\ \overrightarrow{\text{PQ}}$, where P ia any point.
→
Evaluate the following integrals:$\int\frac{\text{x}^2+\text{x}-1}{\text{x}^2+\text{x}-6}\text{ dx}$
→
Solve the following systems of linear equations by cramer's rule:
2x - 3y - 4z = 29,
-2x + 5y - z = -15,
3x - y + 5z = -11
→
If $\text{A}=\begin{bmatrix}3&-5\\-4&2\end{bmatrix},$ then find $A^2 - 5A - 14I$. Hence, obtain $A^3$.
→
Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:
  1. An injective map from A to B.
  2. A mapping from A to B which is not injective.
  3. A mapping from A to B.
→
Find the maximum and the minimum values, if any, without using derivaives of the following functions:$f(x) = 2x^3+ 5$ on $R$.
→
Solve the following differential equation
$\cos\text{x }\frac{\text{dy}}{\text{dx}}-\cos2\text{x}=\cos3\text{x}$
→
find the area of the region included between the parabola $y^2 = x$ and the line $x + y = 2$.
→
Form the differential equation corresponding to $(\text{x}-\text{a})^2+(\text{y}-\text{b})^2=\text{r}^2$ by eliminating a and b.
→
Examine the continuity of the function
$\text{f}\text{(x)}=\begin{cases}3\text{x}-2 &, \text{ if x} \leq 0\\\text{x}+1 &, \text{ if x} > 0\end{cases}\text{at x}=0$
Also sketch the graph of this function.
→