We are asked to find the area of the region \( \left\{ (x, y) : x^2 \leq y \leq |x^2 - 4|, y \geq 1 \right\} \). The given region involves two curves: \( y = x^2 \) and \( y = |x^2 - 4| \). 1. Step 1: Understanding the curves: The curve \( y = x^2 \) is a standard parabola opening upwards. The curve \( y = |x^2 - 4| \) represents two different cases based on the value of \( x \): When \( x^2 \geq 4 \), \( y = x^2 - 4 \). When \( x^2 < 4 \), \( y = 4 - x^2 \). 2. Step 2: Setting up the area integral: The total area can be calculated by integrating the difference between the two curves. The required area is: \[ \text{Area} = 2 \left( \int_{1}^{2} \sqrt{y} \, dy + \int_{2}^{4} (4 - y) \, dy \right) \] 3. Step 3: Calculating the integrals: First integral: \[ \int_{1}^{2} \sqrt{y} \, dy = \left[ \frac{2}{3} y^{3/2} \right]_{1}^{2} = \frac{2}{3} \left( 2^{3/2} - 1 \right) \] Second integral: \[ \int_{2}^{4} (4 - y) \, dy = \left[ 4y - \frac{y^2}{2} \right]_{2}^{4} = \frac{4}{3} \left( 4\sqrt{2} - 1 \right) \]
