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
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