Definite Integration
Leibniz Rule / Limit of Integral
Grade 12
Question:
<p>The value of \(\lim_{h \to 0} \frac{1}{h} \int_{1}^{1+2h} e^{\sqrt{x}} \sin\left(\frac{\pi x}{3}\right) dx\) equals:</p>
<p>(a) \(\sin\dfrac{\pi}{3}\)</p>
<p>(b) \(4e\sin\dfrac{\pi}{3}\)</p>
<p>(c) \(e\sin\dfrac{\pi}{3}\)</p>
<p>(d) \(2e\sin\dfrac{\pi}{3}\)</p>
Step-by-Step Solution
Key Concept: Recognize this limit as the derivative of an integral (Leibniz rule). The expression 1/h × ∫[1 to 1+2h] f(x)dx equals 2f(1) as h→0, since the integral over a small interval [1, 1+2h] of length 2h divided by h gives twice the function value at the left endpoint.
<p><strong>Step 1:</strong> Recognize the form. We have lim(h→0) [1/h]∫[1 to 1+2h] f(x)dx where f(x) = e^√x sin(πx/3).</p><p><strong>Step 2:</strong> By Leibniz rule for differentiating under the integral sign (or equivalently, the mean value theorem for integrals), as h→0:</p><p>lim(h→0) [1/h]∫[a to a+kh] f(x)dx = k·f(a)</p><p>Here a=1, k=2, so the limit = 2·f(1).</p><p><strong>Step 3:</strong> Evaluate f(1) = e^√1 · sin(π·1/3) = e^1 · sin(π/3) = e · (√3/2)</p><p><strong>Step 4:</strong> Therefore, the answer = 2 · e · (√3/2) = e√3</p><p>∴ Answer: <strong>e√3</strong></p>
Correct Answer: D