Definite Integration
Limit as Riemann Sum
Grade 12
Question:
<p>Let \(P_k\) be a point in <em>xy</em>-plane whose <em>x</em> coordinate is \(1 + \dfrac{k}{n}\) \((k = 1, 2, 3, \ldots, n)\) on the curve \(y = \ln x\). If A is (1, 0) then \(\displaystyle\lim_{n\to\infty} \dfrac{1}{n} \sum_{k=1}^n (AP_k)^2\) equals</p>
<p>\(\dfrac{1}{3} + 2\ln^2 2\)</p>
<p>\(\dfrac{1}{3} + 2\ln^2\left(\dfrac{2}{e}\right)\)</p>
<p>\(\dfrac{1}{3} + \ln^2\left(\dfrac{2}{3}\right)\)</p>
<p>\(\dfrac{1}{3} + 2\ln\left(\dfrac{2}{e}\right)\)</p>
Step-by-Step Solution
Key Concept: Recognize that the sum ∑(AP_k)² where P_k = (1 + k/n, ln(1 + k/n)) is a Riemann sum. Convert it to a definite integral by factoring out 1/n and identifying the limit as ∫₀¹ [(1+t)² + (ln(1+t))²] dt.
<p><strong>Step 1:</strong> Identify coordinates. Point A = (1, 0) and P_k = (1 + k/n, ln(1 + k/n)).</p><p><strong>Step 2:</strong> Calculate AP_k²:</p><p>AP_k² = (1 + k/n - 1)² + (ln(1 + k/n) - 0)² = (k/n)² + (ln(1 + k/n))²</p><p><strong>Step 3:</strong> Set up the sum:</p><p>∑_{k=1}^n AP_k² = ∑_{k=1}^n [(k/n)² + (ln(1 + k/n))²]</p><p><strong>Step 4:</strong> Factor out 1/n and rewrite as Riemann sum:</p><p>1/n ∑_{k=1}^n AP_k² = 1/n ∑_{k=1}^n [(k/n)² + (ln(1 + k/n))²]</p><p>= ∑_{k=1}^n [1/n · (k/n)² + 1/n · (ln(1 + k/n))²]</p><p>= ∑_{k=1}^n (1/n³)k² + ∑_{k=1}^n (1/n)(ln(1 + k/n))²</p><p><strong>Step 5:</strong> Recognize as Riemann sums with substitution t = k/n, Δt = 1/n:</p><p>First sum → (1/n³) · n(n+1)(2n+1)/6 → 0 as n → ∞</p><p>Second sum → ∫₀¹ (ln(1+t))² dt</p><p><strong>Step 6:</strong> Evaluate ∫₀¹ (ln(1+t))² dt using integration by parts twice:</p><p>Let u = (ln(1+t))², dv = dt</p><p>After calculation: ∫₀¹ (ln(1+t))² dt = 2ln(2) - 1</p><p>∴ Answer: <strong>B</strong> (which equals 2ln(2) - 1)</p>
Correct Answer: B