Definite Integration
Properties of Definite Integrals
Grade 12
Question:
<p>If <i>f</i> and <i>g</i> are continuous functions in <i>[0,1]</i> satisfying <i>f(x) = f(a - x)</i> and <i>g(x) + g(a - x) = a</i>, then <i>∫₀ᵃ f(x) × g(x) dx</i> is equal to</p>
<p>(a) <i>a</i>/2 ∫₀ᵃ f(x) dx</p>
<p>(b) ∫₀ᵃ f(x) dx</p>
<p>(c) 0</p>
<p>(d) <i>a</i> ∫₀ᵃ f(x) dx</p>
Step-by-Step Solution
Key Concept: Use symmetry properties of f and the constraint on g to simplify the integral through substitution.
<p>Using the properties of symmetry: Since f(x) = f(a-x) and g(x) + g(a-x) = a, we can apply substitution and symmetry arguments to show that ∫₀ᵃ f(x) × g(x) dx = (a/2)∫₀ᵃ f(x) dx.</p>
Correct Answer: A