Question

The area of the region bounded by the curve \( y = |x - 2| \) between \( x = 1 \) and \( x = 3 \).

• A. 1 sq. units
• B. 2 sq. units
• C. 3 sq. units
• D. 4 sq. units

Hint:

When dealing with absolute value functions, break the integral into cases based on the behavior of the function. This allows you to handle each section separately and properly calculate the area.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to find the area under the absolute value function \( y = |x - 2| \) from \( x=1 \) to \( x=3 \).
Step 2: Key Formula or Approach:
The area is given by the definite integral \( \text{Area} = \int_{1}^{3} |x - 2| \text{ d}x \).
Step 3: Detailed Explanation:
1. The function \( |x - 2| \) behaves differently around its vertex at \( x = 2 \).
- For \( 1 \le x<2 \), \( |x - 2| = -(x - 2) = 2 - x \).
- For \( 2 \le x \le 3 \), \( |x - 2| = x - 2 \).
2. Split the integral:
\( \text{Area} = \int_{1}^{2} (2 - x) \text{ d}x + \int_{2}^{3} (x - 2) \text{ d}x \)
\( \text{Area} = \left[ 2x - \frac{x^2}{2} \right]_{1}^{2} + \left[ \frac{x^2}{2} - 2x \right]_{2}^{3} \)
\( \text{Area} = ((4 - 2) - (2 - 1/2)) + ((9/2 - 6) - (2 - 4)) \)
\( \text{Area} = (2 - 3/2) + (-3/2 + 2) \)
\( \text{Area} = 1/2 + 1/2 = 1 \).
Alternative Geometric Method:
The region consists of two right-angled triangles with base 1 and height 1.
\( \text{Area} = 2 \times (1/2 \times 1 \times 1) = 1 \).
Step 4: Final Answer:
The area is 1 sq. units.