Definite Integration
Riemann Sums and Limits
Grade 12
Question:
<p>If \(f(\theta) = \lim_{n \to \infty} \sum_{r=0}^{n\theta} \frac{1}{n(3\theta n - 2r)(n\theta + 2r)}\), then:</p>
<p>(a) \(f(1) = \frac{\pi}{6}\)</p>
<p>(b) \(f(\theta) = \frac{1}{2} \int_0^q \frac{dx}{q^2 - (x - q/2)^2}\)</p>
<p>(c) \(f(\theta)\) is a constant function</p>
<p>(d) \(y = f(\theta)\) is invertible</p>
Step-by-Step Solution
Key Concept: Recognize Riemann sums and convert them to definite integrals.
<p>Convert the Riemann sum to a definite integral. The sum represents $\int_0^\theta \frac{d\theta}{(3\theta - 2r/n)(\theta + 2r/n)}$ in the limit, which equals the given integral form.</p>
Correct Answer: b