Definite Integration
Estimation of Definite Integrals
Grade 12
Question:
<p>The \(L\) denotes the value of the definite integral \(\displaystyle\int_0^1 \dfrac{1}{1+x^8}\,dx\), then which one of the following must be true?</p>
<p>(a) \(\dfrac{\pi}{4} < L < 1\)</p>
<p>(b) \(L = \dfrac{\pi}{4}\)</p>
<p>(c) \(L > 1\)</p>
<p>(d) \(0 < L < \dfrac{\pi}{4}\)</p>
Step-by-Step Solution
Key Concept: Establish bounds for the integral by comparing the integrand with simpler functions whose integrals are known, using the fact that 0 ≤ 1/(1+x⁸) ≤ 1 on [0,1].
<p><strong>Step 1:</strong> Establish the domain constraints. For x ∈ [0,1], we have 0 ≤ x⁸ ≤ 1, so 1 ≤ 1+x⁸ ≤ 2.</p><p><strong>Step 2:</strong> Apply reciprocal inequality (reversing inequality for positive numbers): 1/(1+x⁸) ≤ 1, and also 1/(1+x⁸) ≥ 1/2.</p><p><strong>Step 3:</strong> Integrate across [0,1]: ∫₀¹ (1/2) dx ≤ ∫₀¹ 1/(1+x⁸) dx ≤ ∫₀¹ 1 dx</p><p><strong>Step 4:</strong> Evaluate boundary integrals: 1/2 ≤ L ≤ 1</p><p>∴ L must lie in the interval [1/2, 1]</p>
Correct Answer: A