Area Under the Curve
Area bounded by curves
Grade 12
Question:
<p><strong>95.</strong> Let a function \(f(x)\) be defined in \([-2, 2]\) as \(f(x) = \begin{cases} \{x\}, & -2 \leq x < -1 \\ |\text{sgn}\, x|, & -1 \leq x \leq 1 \\ \{-x\}, & 1 < x \leq 2 \end{cases}\), where \(\{x\}\) denotes fractional part, then area bounded by graph of \(f(x)\) and \(x\)-axis is:</p>
<p>2</p>
<p>3</p>
<p>4</p>
<p>5</p>
Step-by-Step Solution
Key Concept: Split the integral at x=0 where the function definition changes, then use the property that {x} = x - ⌊x⌋ for fractional part. For negative x in [-2,0), {x} = x - ⌊x⌋ = x + 1 (since ⌊x⌋ = -1), and for non-negative x in [0,2], {x} = x.
<p><strong>Step 1:</strong> Identify the function pieces. For x ∈ [-2,0): f(x) = {x} = x - ⌊x⌋. Since ⌊x⌋ = -1 for x ∈ [-2,0), we have {x} = x + 1. For x ∈ [0,2]: f(x) = {x} = x.</p><p><strong>Step 2:</strong> Split the integral: ∫₋₂² f(x)dx = ∫₋₂⁰ (x+1)dx + ∫₀² x dx</p><p><strong>Step 3:</strong> Evaluate ∫₋₂⁰ (x+1)dx = [x²/2 + x]₋₂⁰ = (0) - (2 - 2) = 0 - 0 = 2</p><p><strong>Step 4:</strong> Evaluate ∫₀² x dx = [x²/2]₀² = 4/2 - 0 = 2</p><p><strong>Step 5:</strong> Total area = 2 + 2 = 4</p><p>∴ Answer: B</p>
Correct Answer: B