Step 1: Understanding the Concept:
The absolute value function \( y = |x-2| \) represents a V-shaped graph with a vertex at \( (2,0) \). The area between \( x=1 \) and \( x=3 \) is composed of two identical triangles.
Step 2: Key Formula or Approach:
Area \( = \int_{x_1}^{x_2} y \, dx \).
Since the function has a cusp at \( x=2 \), we split the integral:
\[ \text{Area} = \int_{1}^{2} |x - 2| dx + \int_{2}^{3} |x - 2| dx \]
Step 3: Detailed Explanation:
In the interval \( [1, 2] \), \( |x-2| = -(x-2) = 2-x \).
\[ A_1 = \int_{1}^{2} (2-x) dx = [2x - \frac{x^2}{2}]_1^2 = (4 - 2) - (2 - 0.5) = 2 - 1.5 = 0.5 \]
In the interval \( [2, 3] \), \( |x-2| = x-2 \).
\[ A_2 = \int_{2}^{3} (x-2) dx = [\frac{x^2}{2} - 2x]_2^3 = (4.5 - 6) - (2 - 4) = -1.5 - (-2) = 0.5 \]
Total Area \( = A_1 + A_2 = 0.5 + 0.5 = 1 \) sq unit.
Step 4: Final Answer:
The area is 1 sq unit.