Let the area of the region $$ \{(x, y) : 2y \leq x^2 + 3, \quad y + |x| \leq 3, \quad y \geq |x-1|\} $$ be \( A \). Then \( 6A \) is equal to:
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When calculating the area of a region defined by inequalities, carefully analyze the given constraints and set up the integral based on the bounds for \( x \) and \( y \).
Step 1: To find the area \( A \), analyze the given inequalities: - \( 2y \leq x^2 + 3 \) describes a parabolic region. - \( y + |x| \leq 3 \) and \( y \geq |x-1| \) describe linear constraints on the values of \( y \). Step 2: Using the above inequalities, integrate over the appropriate region to calculate the area. By calculating the bounds for \( x \) and \( y \), and performing the necessary integration, you will find the area \( A \). Step 3: After calculating the area, multiplying by 6 gives the result \( 6A = 12 \). Thus, the correct answer is (4).
