Question

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: 

• A. 16
• B. 18
• C. 14
• D. 12

Hint:

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 \( y \).

Answer 1

The Correct Option is D

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).