Step 1: Understanding the Concept:
The absolute value function \( |x - a| \) is a piecewise function defined as:
\( |x - a| = (x - a) \) for \( x \geq a \)
\( |x - a| = -(x - a) = (a - x) \) for \( x<a \)
For the integral from 0 to 2, we need to check how \( x \) behaves relative to the critical point \( x = 2 \).
Step 2: Key Formula or Approach:
Identify the interval: \( 0 \leq x \leq 2 \).
In this entire interval, \( x \) is less than or equal to 2.
Therefore, \( x - 2 \leq 0 \).
By the definition of absolute value, \( |x - 2| = -(x - 2) = 2 - x \).
Step 3: Detailed Explanation:
Rewrite the integral using the definition found in Step 2:
\[ I = \int_{0}^{2} (2 - x) dx \]
Now, integrate the expression term by term using the power rule:
\[ I = \int_{0}^{2} 2 dx - \int_{0}^{2} x dx \]
\[ I = [2x]_{0}^{2} - \left[ \frac{x^{2}}{2} \right]_{0}^{2} \]
Evaluating at the boundaries:
\[ I = (2(2) - 2(0)) - \left( \frac{2^{2}}{2} - \frac{0^{2}}{2} \right) \]
\[ I = 4 - \left( \frac{4}{2} \right) \]
\[ I = 4 - 2 = 2 \]
Geometric Method:
The graph of \( y = |x - 2| \) from \( x = 0 \) to \( x = 2 \) is a straight line from \( (0, 2) \) to \( (2, 0) \).
This forms a right-angled triangle with base = 2 and height = 2.
Area = \( \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times 2 = 2 \).
Both methods confirm the same result.
Step 4: Final Answer:
The value of the integral is 2.