Question

The area bounded by \( y - 1 = -|x| \) and \( y + 1 = |x| \) is:

• A. \( \frac{1}{2} \)
• B. \( 1 \)
• C. \( 2 \)
• D. \( 0 \)

Hint:

The area bounded by two curves \( f(x) \) and \( g(x) \) from \( a \) to \( b \) is \( \int_{a}^{b} |f(x) - g(x)| dx \). Consider symmetry to simplify the integration.

Answer 1

The Correct Option is C

Step 1: Rewrite the equations.
\( y = 1 - |x| \) and \( y = |x| - 1 \).
Step 2: Find intersection points.
Setting \( 1 - |x| = |x| - 1 \implies 2 = 2|x| \implies |x| = 1 \implies x = \pm 1 \). Points are \( (-1, 0) \) and \( (1, 0) \).
Step 3: Set up the integral for the area.
Area \( = \int_{-1}^{1} [(1 - |x|) - (|x| - 1)] dx = \int_{-1}^{1} (2 - 2|x|) dx \).
Step 4: Evaluate the integral using symmetry.
Area \( = 2 \int_{0}^{1} (2 - 2x) dx = 2 [2x - x^2]_{0}^{1} = 2 [(2 - 1) - (0)] = 2 \).