Mathematics

# Evaluate the following integral as limit of sum:$\displaystyle \int_{a}^{b}x\ dx$

##### SOLUTION
Let $I=\displaystyle \int_a^b\ x \ dx$

We know that

$\displaystyle \lim_{h\to 0} [f(a)+f(a+h)+f(a+2h)+....+f(a+(n-1)h)]$, where $h=\dfrac {b-a}{n}$

Here, $f(x)=x$

$\therefore \displaystyle \int_{a}^b = \displaystyle \lim_{x\to 0} h[a+(a+h)+(a+2h)+...+(a+(n-1)h]$

$\Rightarrow \ \displaystyle \int_a^b x\ dx \displaystyle \lim_{x\to 0} [na+\left\{h+2h+3h+...+(n-1)h \right\}]$

$\Rightarrow \ \displaystyle \int_a^b x\ dx \displaystyle \lim_{x\to 0} [nha +h^2 \dfrac {n(n-1)}{2} =\displaystyle \lim_{x\to 0} [(nh) a+ (nh) \dfrac {(nh-h)}{2}]]$

$\Rightarrow \ \displaystyle \int_a^b x\ dx \displaystyle \lim_{x\to 0} [ (b-a) a+ \dfrac {(b-a)(b-a-h)}{2} ]=(b-a) a+\dfrac {(b-a)^2}{2}=\dfrac {b^2 -a^2}{2}$

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