using-the-limit-defination-find-the-area-of-f-x-cos-x-0-pi-2-

Question Number 31141 by Cheyboy last updated on 03/Mar/18

u s i n g t h e l i m i t d e f i n a t i o n

f i n d t h e a r e a o f

f (x) = c o s (x) [0, π / 2]

Answered by Joel578 last updated on 03/Mar/18

A = \int_{0}^{\frac{π}{2}} \cos x d x = \lim_{n \to \infty} \underset{i = 1}{\sum^{n}} f (x_{i}) Δ x_{i}

= \lim_{n \to \infty} \underset{i = 1}{\sum^{n}} \cos (\frac{i π}{2 n}) \frac{π}{2 n}

= \lim_{n \to \infty} \frac{π}{2 n} \underset{i = 1}{\sum^{n}} \cos (\frac{i π}{2 n})

= \frac{π}{2} \lim_{n \to \infty} \frac{\underset{i = 1}{\sum^{n}} \cos (\frac{i π}{2 n})}{n}

= \frac{π}{2} \lim_{n \to \infty} \frac{\cos (\frac{π}{2 n}) + \cos (\frac{π}{n}) + \cos (\frac{3 π}{2 n}) + \dots + \cos (\frac{n π}{2 n})}{n}

Facebook Tweet Pin