Mathematics

$\displaystyle \int_0^{\pi} sin x =\displaystyle \lim_{n \rightarrow \infty} \displaystyle \Sigma_{i=1}^n sin \left(\dfrac{\pi i}{n}\right) \dfrac{\pi}{n}$State whether the above statement is True or False?

True

SOLUTION
We have to state whether $\int_{0}^{\pi} sin x dx$ can be represented as $\lim_{n \rightarrow \infty} \sum_{i=1}^{n} sin \left(\dfrac{\pi i}{n}\right)\dfrac{\pi}{n}$ is true or false.

We know that $\int_{a}^{b} f(x) dx=\lim_{n \rightarrow \infty} \left[\dfrac{b-a}{n} \sum_{i=1}^{n} f\left(a+i\dfrac{b-a}{n}\right)\right]$

Here we have $f(x)=sin x,a=0,b=\pi$

Therefore $\int_{0}^{\pi} sin x dx=\lim_{n \rightarrow \infty} \left[\dfrac{\pi-0}{n} \sum_{i=1}^{n} f\left(0+i\dfrac{\pi-0}{n}\right)\right]$

$=\lim_{n \rightarrow \infty} \left[\dfrac{\pi}{n} \sum_{i=1}^{n} f\left(\dfrac{\pi i}{n}\right)\right]$

$=\lim_{n \rightarrow \infty} \left[\dfrac{\pi}{n} \sum_{i=1}^{n} sin \left(\dfrac{\pi i}{n}\right)\right]$

$\int_{0}^{\pi} sin x dx=\lim_{n \rightarrow \infty} \sum_{i=1}^{n} sin \left(\dfrac{\pi i}{n}\right)\dfrac{\pi}{n}$

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TRUE/FALSE Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

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