<p>The area of region \(R\) that is completely bounded by the graph of \(f(x) = 2x - 1\) and \(g(x) = x^2 - 4\) is ______ (up to two decimal places).</p>
Step-by-Step Solution
Key Concept: Find intersection points of the two curves, then integrate the absolute difference of the upper and lower functions over the bounded interval. The region's area equals the integral of |f(x) - g(x)| between intersection points.
<p><strong>Step 1: Find intersection points</strong><br/>Set f(x) = g(x):<br/>2x - 1 = x² - 4<br/>x² - 2x - 3 = 0<br/>(x - 3)(x + 1) = 0<br/>x = 3 or x = -1</p><p><strong>Step 2: Determine which function is above</strong><br/>Test x = 0: f(0) = -1, g(0) = -4<br/>Since -1 > -4, we have f(x) > g(x) on [-1, 3]</p><p><strong>Step 3: Set up the integral</strong><br/>Area = ∫₋₁³ [f(x) - g(x)] dx<br/>= ∫₋₁³ [(2x - 1) - (x² - 4)] dx<br/>= ∫₋₁³ (-x² + 2x + 3) dx</p><p><strong>Step 4: Evaluate the integral</strong><br/>= [-x³/3 + x² + 3x]₋₁³<br/>= [(-27/3 + 9 + 9) - (1/3 + 1 - 3)]<br/>= [18 - (-5/3)]<br/>= 18 + 5/3<br/>= 54/3 + 5/3<br/>= 59/3</p><p><strong>∴ Answer: 19.67</strong> (or 59/3 ≈ 19.67)</p>
Correct Answer: 19