AP EAMCET · Maths · Definite Integration
\(\lim _{n \rightarrow \infty} \frac{\pi}{2 n}\left[\sin \frac{\pi}{2 n}+\sin \frac{2 \pi}{2 n}+\sin \frac{3 \pi}{2 n}+\ldots+\sin \frac{\pi}{2}\right]=\)
- A \(1\)
- B \(0\)
- C \(4\)
- D \(3\)
Answer & Solution
Correct Answer
(A) \(1\)
Step-by-step Solution
Detailed explanation
\( \lim _{n \rightarrow \infty} \sum_{k=1}^{n} \sin\left(\frac{k\pi}{2n}\right) \frac{\pi}{2 n} = \int_{0}^{\frac{\pi}{2}} \sin(x) dx \) \( = [-\cos(x)]_{0}^{\frac{\pi}{2}} \) \( = -\cos\left(\frac{\pi}{2}\right) - (-\cos(0)) \) \( = -0 + 1 \) \( = 1 \)
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