Evaluate the following integrals as limit of sum: ^ b _ a x dx - Vidyadip

Question

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

✓

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

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

Solve the following differential equation
$\sin^4\text{x}\frac{\text{dy}}{\text{dx}}=\cos\text{x}$

There are 2 families A and B. There are 4 men, 6 women and 2 children in family A, and 2 men, 2 women and 4 children in family B. The recommended daily amount of calories is 2400 for men, 1900 for women, 1800 for children and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent the above information using matrices. Using matrix multiplication, calculate the total requirement of calories and proteins for each of the 2 families. What awareness can you create among people about the balanced diet from this question?

Show that the vectors $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ given by $\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}},\ \vec{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ are non-coplanar. Express vector $\vec{\text{d}}=2\hat{\text{i}}-\hat{\text{j}}-3\hat{\text{k}}$ as a linear combination of the vectors $\vec{\text{a}},\ \vec{\text{b}}\text{ and }\vec{\text{c}}$.

Prove that the function $f : R → R$ defined by $f (x) = 2x + 5$ is one-one.

A wholesale dealer deals in two kinds, $A$ and $B$ (say) of mixture of nuts. Each kg of mixture $A$ contains $60$ grams of almonds, $30$ grams of cashew nuts and $30$ grams of hazel nuts. Each kg of mixture $B$ contains $30$ grams of almonds, $60$ grams of cashew nuts and $180$ grams of hazel nuts. The remainder of both mixtures is per nuts. The dealer is contemplating to use mixtures $A$ and $B$ to make a bag which will contain at least $240$ grams of almonds, $300$ grams of cashew nuts and $540$ grams of hazel nuts. Mixture $A$ costs Rs. $8$ per kg. and mixture $B$ costs Rs. $12$ per kg. Assuming that mixtures A and B are uniform, use graphical method to determine the number of kg . of each mixture which he should use to minimise the cost of the bag.

Using properties of determinants prove that:
$\begin{vmatrix}\text{x}+4&2\text{x}&2\text{x}\\2\text{x}&\text{x}+4&2\text{x}\\2\text{x}&2\text{x}&\text{x}+4\end{vmatrix}$
$=(5\text{x}+4)(4-\text{x})^2$

If a unit vector $\vec{\text{a}}$ makes an angle $\frac{\pi}3$ with $\hat{\text{i}}$, $\frac{\pi}4$ with $\hat{\text{j}}$ and an acute angle $\theta$ with $\hat{\text{k}}$, then find $\theta$ and hence, the components of $\vec{\text{a}}$.

Find the shortest distance between the lines $\frac{{x + 1}}{7} = \frac{{y + 1}}{{ - 6}} = \frac{{z + 1}}{1}$ and $\frac{{x - 3}}{1} = \frac{{y - 5}}{{ - 2}} = \frac{{z - 7}}{1}$

$\text{Evaluate}: \int\limits^{\pi}_{-\pi} (\cos ax - \sin bx)^{2} dx$

Use product $\begin{bmatrix}1&-1&2\\0&2&-3\\3&-2&4\end{bmatrix}\begin{bmatrix}-2&0&1\\9&2&-3\\6&1&-2\end{bmatrix}$ to solve the system of equations $x + 3z = 9, -x + 2y - 2z = 4, 2x - 3y + 4z = -3.$