Which of the following regions will represent the shaded area in the given figure ?

1. Identify the Core Equations from the Figure:

• The Parabola: The label says y 5 x^2 1 2, which translates to the quadratic curve: $y = x^2 + 2$ (with its vertex at $(0,2)$).
• The Straight Line: The label reads y 5 2x 1 2, which translates to the linear equation: $y = 2x + 2$.
• Vertical Boundary Lines: There are two vertical lines drawn at x 5 2 and x 5 3, meaning $x = 2$ and $x = 3$.

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2. Analyze the Shaded Region:

Let's look at where the shaded area lies relative to each boundary:

1. With respect to the X-axis ($y = 0$): The shaded region lies entirely above the x-axis, meaning $y \geq 0$.
2. With respect to the Parabola ($y = x^2 + 2$): The shaded area lies below the curve of the parabola for its entire span. Therefore, the inequality must be $y \leq x^2 + 2$.
3. With respect to the Line ($y = 2x + 2$): The shaded area lies below the straight line. Therefore, the inequality must be $y \leq 2x + 2$.
4. With respect to the X-interval: The region starts at the y-axis ($x = 0$) and extends all the way to the final vertical line boundary at $x = 3$. This means $0 \leq x \leq 3$.

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Conclusion:

Combining all these boundaries, the system of inequalities that represents the shaded region is:

$\{(x,y) : 0 \le y \le x^2+2, \ 0 \le y \le 2x+2, \ 0 \le x \le 3\}$

This corresponds exactly to Option 1.