Download App

Definite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals5 Marks
Question
Evaluate the following integrals as limit of sum:
$\int\limits^{3}_{2}\text{x}^2\text{ dx}$
✓

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, $\text{a}=2,\text{ b}=3,\text{ f(x)}=\text{x}^2,\text{ h}=\frac{3-2}{\text{n}}=\frac{1}{\text{n}}$
Therefore, $\text{I}=\int\limits^{3}_{2}\text{x}^2\text{ dx}$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\big[\text{f}(2)+\text{f}(2+\text{h})\ ....\ +\text{f}\big\{2+(\text{n}-1)\text{h}\big\}\big]$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\Big[2^2+(2+\text{h})^2+\ ....+\ \big\{2(\text{n}-1)\text{h}\big\}^2\Big]$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\Big[4\text{n}+\text{h}\big\{1^2+2^2+3^2+\ ....\ +(\text{n}-1)^2+4\text{h}\big\}\\+4\text{h}\big\{1+2+\ ....+\ (\text{n}-1)\text{h}\Big]$
$=\lim\limits_{\text{h}\rightarrow0}\text{h}\Big[4\text{n}+\text{h}^2\frac{\text{n}(\text{n}-1)(\text{n}-2)}{6}+4\text{h}\frac{\text{n}(\text{n}-1)}{2}\Big]$
$=\lim\limits_{\text{h}\rightarrow\infty}\frac{1}{\text{n}}\Big[4\text{n}+\frac{(\text{n}-1)(2\text{n}-1)}{6\text{n}}+2\text{n}-2\Big]$
$=\lim\limits_{\text{h}\rightarrow\infty}5\Big[6+\frac{1}{6}\Big(1-\frac{1}{\text{n}}\Big)\Big(2-\frac{1}{\text{n}}\Big)-\frac{2}{\text{n}}\Big]$
$=6+\frac{1}{3}$
$=\frac{19}{3}$

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 "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 $\text{f}\text{(x)}=\begin{cases}\frac{1-\cos\text{x}}{\text {x}^2}, & \text{when} \text{ x}\neq 0\\1, & \text{when}\text{ x} = 0\end{cases}$ Show that f(x) is discontinuous at x = 0.
→
If a white ball is selected, what is the probability that it came from:
  1. Bag 2
  2. Bag 3
→
Find the particular solution of the following differential equation;
$\frac{\text{dx}}{\text{dy}} = 1 + x^2 + y^2 + x^2y^2, $ given that $y = 1$ when $x = 0$.
→
Evaluate the following intregals:
$\int\frac{2}{2+\sin^22\text{x}}\text{ dx}$
→
Write a value of $\int\text{e}^{\log\sin\text{x}}\cos\text{x}\text{ dx}$
→
It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 5 mice are inoculated, find the probability that.
  1. none contract the disease.
  2. more than 3 contract the disease.
→
If $\text{y}=\cot^{-1}\Big\{\frac{\sqrt{1+\sin\text{x}}+\sqrt{1-\sin\text{x}}}{\sqrt{1+\sin\text{x}}-\sqrt{1-\sin\theta}}\Big\},$ show that $\frac{\text{dy}}{\text{dx}}$ is independent of x.
→
Using properties of determinants, prove that:
$\begin{vmatrix}\alpha&\alpha^2&\beta+\gamma\\\beta&\beta^2&\gamma+\alpha\\\gamma&\gamma^2&\alpha+\beta\end{vmatrix}=(\beta-\gamma)(\gamma-\alpha)(\alpha-\beta)(\alpha+\beta+\gamma)$
→
Find the equation of the curve passing through the point (0, 1) if the slope of the tangent to the curve at each of its point is equal to the sum of the abscissa and the product of the abscissa and the ordinate of the point.
→
Evaluate $\int\limits_0^{3}(2x^2 + 3x + 5)dx$ as limit of a sum.
→