Mathematics

# Evaluate the following integral as limit of sum:$\displaystyle \int_{1}^{4}(x^2-x)dx$

##### SOLUTION
We know that
$\displaystyle \int_a^b f(x) dx= \displaystyle \lim _{h\to 0} h[f(a)+f(a+h)+f(a+2h) +....+f(a+(n-1)h)]$
Here $a=1, b=4, h=\dfrac {3}{n}$ and $f(x) =x^2 -x$
$\therefore \ I=\displaystyle \int_{1}^4 (x^2-x) dx \displaystyle \lim_{h\to 0} h[ f(1) + f(1+h)+f(1-2h) +...+f(1+(n-1) h)]$
$\Rightarrow =I\displaystyle \lim _{ h\rightarrow 0 }{ h\left[ \left\{ \left( { 1 }^{ 2 }-1 \right) \right\} +\left\{ { \left( 1+h \right) }^{ 2 }-\left( 1+h \right) \right\} +\left\{ { \left( 1+2h \right) }^{ 2 }-\left( 1+2h \right) \right\} +...+\left\{ { \left( 1+\left( n-1 \right) h \right) }^{ 2 }-\left( 1+\left( n-1 \right) h \right) \right\} \right] }$
$\Rightarrow =I\displaystyle \lim _{ h\rightarrow 0 }{ h\left[ \left\{ { 1 }^{ 2 }+{ \left( 1+h \right) }^{ 2 }+...+{ \left( 1+\left( n-1 \right) h \right) }^{ 2 } \right\} -\left\{ 1+\left( 1+h \right) +\left( 1+2h \right) +....+\left( 1+\left( n-1 \right) h \right) \right\} \right] }$
$\Rightarrow =I\displaystyle \lim _{ h\rightarrow 0 }{ h\left[ \left\{ 1+2h\left( 1+2+3+...+\left( n-1 \right) h \right) +{ h }^{ 2 }\left( { 1 }^{ 2 }+{ 2 }^{ 2 }+....+{ \left( n-1 \right) }^{ 2 } \right) \right\} \left\{ n+h\left( 1+2+3+...+\left( n-1 \right) \right) \right\} \right] }$
$\Rightarrow =I\displaystyle \lim _{ h\rightarrow 0 }{ h\left[ \left\{ n+hn\left( n-1 \right) +{ h }^{ 2 }\frac { n\left( n-1 \right) \left( 2n-1 \right) }{ 6 } \right\} -\left\{ n+h\frac { n\left( n-1 \right) }{ 2 } \right\} \right] }$
$\Rightarrow \ I=\displaystyle \lim_{h\to 0}h \left [h\dfrac {(n-1)}{2}+h^2 \dfrac {n(n-1) (2n-1)}{n^2}\right]$
$\Rightarrow \ I=\displaystyle \lim_{h\to \infty} \left\{\dfrac {9}{2} \left (\dfrac {n-1}{n}\right) +\dfrac {27}{6} \dfrac {(n-1)(2n-1)}{n^2}\right\}$
$\Rightarrow \ I=\displaystyle \lim_{h\to \infty} \left\{\dfrac {9}{2} \left(1-\dfrac {1}{n}\right) +\dfrac {9}{2} \left (2-\dfrac {1}{n}\right) \right\} =\dfrac {9}{2} +9=\dfrac {27}{2}$

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Hard
The value of $\displaystyle \int _{ -1 }^{ 1 } max\left\{ { \: { 2-x,2,1+x } } \right\} dx$ is?
• A. $4$
• B. $2$
• C. none of these
• D. $\displaystyle \frac{9}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate $\displaystyle \int e^x(x^2+2x) d x$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate the following integral
$\int { \cfrac { \cos { x } }{ \cos { \left( x-a \right) } } } dx\quad$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
For $n > 0,$ solution of integral $\displaystyle \int_{0}^{2\pi }\frac{x\sin ^{2n}x}{\sin ^{2n}x+\cos ^{2n}x}dx$, is equal to
• A. $\dfrac{\pi}2$
• B. $2\pi$
• C. $\dfrac{\pi}{4}$
• D. $\pi^2$

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.