<p>Given the region \(\dfrac{y^2}{2} \le x \le y + 4\), find the area of the region (in square units).</p>
Step-by-Step Solution
Key Concept: Identify the parabola and line boundaries, then determine which curve is the right boundary by testing intersection points and regions. Integration with respect to y eliminates the need to split the region into multiple pieces.
<p><strong>Step 1:</strong> Identify the boundaries. We have a parabola <strong>x = y²/2</strong> and a line <strong>x = y + 4</strong>.</p><p><strong>Step 2:</strong> Find intersection points by setting y²/2 = y + 4:<br/>y² - 2y - 8 = 0<br/>(y - 4)(y + 2) = 0<br/>y = 4 or y = -2<br/>Points: (-1, -2) and (8, 4)</p><p><strong>Step 3:</strong> Determine which is the right boundary. At y = 0: parabola gives x = 0, line gives x = 4. So the line is to the right.</p><p><strong>Step 4:</strong> Set up the integral with respect to y:<br/>A = ∫₍₋₂₎⁴ [(y + 4) - y²/2] dy</p><p><strong>Step 5:</strong> Evaluate:<br/>A = [y²/2 + 4y - y³/6]₍₋₂₎⁴<br/>= (8 + 16 - 64/6) - (2 - 8 + 8/6)<br/>= (24 - 32/3) - (-6 + 4/3)<br/>= 24 - 32/3 + 6 - 4/3<br/>= 30 - 36/3 = 30 - 12</p><p>∴ <strong>Answer: 18 square units</strong></p>
Correct Answer: 18