Integral Calculus-2
Integral Calculus-2
Allen Star Batch
Grade 12
Question:
For positive integers $k = 1, 2, 3, \ldots, n$, let $S_k$ denotes the area of $\triangle AOB_k$ (where 'O' is origin) such that $\angle AOB_k = \frac{k\pi}{2n}$, $OA = 1$ and $OB_k = k$. If the value of $\lim_{n \to \infty} \frac{1}{n^2} \sum_{k=1}^n S_k = \frac{a}{\pi^2}$, then 'a' is equal to
Step-by-Step Solution
Key Concept: Riemann sums with careful parameter identification convert discrete geometric approximations into definite integrals.
The area of triangular sectors is approximated using Riemann sums. With $OB_k = k$ and $\angle AOB_k = \frac{k\pi}{2n}$, each sector has area $S_k = \frac{1}{2}k\sin\frac{k\pi}{2n}$. The total area is $L = \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n S_k = \frac{k}{2n^2}\sum_{n=1}^\infty \sin\frac{k\pi}{2n} = \frac{2}{\pi^2}$, obtained by converting the sum to a Riemann integral.
Correct Answer: 2