Question

The value of \( \int_0^2 |x - 2| dx = \)

• A. 1
• B. 2
• C. 3
• D. 3/2

Hint:

For definite integrals involving absolute value functions over simple intervals, drawing a quick sketch of the graph and calculating the area geometrically (e.g., as triangles or rectangles) can be much faster and less error-prone than algebraic integration.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks to evaluate a definite integral involving an absolute value function.

Step 2: Key Formula or Approach:
1. Define the absolute value function: \( |A| = A \) if \( A \ge 0 \), and \( |A| = -A \) if \( A < 0 \).
2. Split the integral into intervals where the expression inside the absolute value changes sign.
3. Evaluate the integral over each interval.
Alternatively, interpret the integral geometrically as the area under the curve.

Step 3: Detailed Explanation:
Given integral: \( I = \int_0^2 |x - 2| dx \)
Analyze the expression inside the absolute value, \( x - 2 \):
- For \( x \le 2 \), \( x - 2 \le 0 \). So, \( |x - 2| = -(x - 2) = 2 - x \).
The interval of integration is from 0 to 2. In this entire interval, \( x \le 2 \).
Therefore, for \( x \in [0, 2] \), \( |x - 2| \) can be replaced by \( 2 - x \).
\[ I = \int_0^2 (2 - x) dx \]
Now, integrate term by term:
\[ I = \left[ 2x - \frac{x^2}{2} \right]_0^2 \]
Evaluate at the limits:
\[ I = \left( 2(2) - \frac{2^2}{2} \right) - \left( 2(0) - \frac{0^2}{2} \right) \]
\[ I = \left( 4 - \frac{4}{2} \right) - (0 - 0) \]
\[ I = (4 - 2) - 0 \]
\[ I = 2 \]
Geometric Interpretation:
The function \( y = |x - 2| \) is a V-shaped graph with its vertex at \( x = 2 \).
For the interval \([0, 2]\), the function is \( y = 2 - x \).
This is a straight line.
- At \( x = 0 \), \( y = 2 \).
- At \( x = 2 \), \( y = 0 \).
The integral represents the area of a right-angled triangle with base 2 (from \( x=0 \) to \( x=2 \)) and height 2 (at \( x=0 \)).
Area = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 2 = 2$.

Step 4: Final Answer:
The value of the integral is 2.