Step 1: Understanding the Concept:
The area under a curve \(y = f(x)\) from \(x=a\) to \(x=b\) is given by the definite integral \(\int_{a}^{b} f(x) dx\). Since the function is an absolute value, we must split the integral at the point where the expression inside the absolute value becomes zero. Step 2: Key Formula or Approach:
Area \(= \int_{1}^{6} |5 - x| dx\).
The function changes behavior at \(x = 5\). Step 3: Detailed Explanation:
1. Split the integral:
\[ \text{Area} = \int_{1}^{5} |5 - x| dx + \int_{5}^{6} |5 - x| dx \]
2. For \(x<5\), \(|5 - x| = 5 - x\). For \(x>5\), \(|5 - x| = x - 5\).
\[ \text{Area} = \int_{1}^{5} (5 - x) dx + \int_{5}^{6} (x - 5) dx \]
3. Integrate:
\[ \text{Area} = \left[ 5x - \frac{x^{2}}{2} \right]_{1}^{5} + \left[ \frac{x^{2}}{2} - 5x \right]_{5}^{6} \]
4. Evaluate limits:
\[ \text{Part 1} = (25 - 12.5) - (5 - 0.5) = 12.5 - 4.5 = 8 \]
\[ \text{Part 2} = (18 - 30) - (12.5 - 25) = -12 + 12.5 = 0.5 \]
5. Total Area:
\[ \text{Total} = 8 + 0.5 = 8.5 = 17/2 \text{ sq units} \] Step 4: Final Answer:
The area is 17/2 sq units.