Question

The area of the region bounded by the curves \( y = |5 - x| \), \( x = 1 \), \( x = 6 \) and the X-axis is:

• A. 15 sq units
• B. \( \frac{17}{2} \) sq units
• C. 13 sq units
• D. 16 sq units

Hint:

For modulus graphs, always split at the point where the expression inside becomes zero.

Answer 1

The Correct Option is B

Concept: For modulus functions, split at the point where the expression inside modulus becomes zero. \[ |5 - x| = \begin{cases} 5 - x, & x \le 5
x - 5, & x>5 \end{cases} \]

Step 1: Split the interval \[ \text{Area} = \int_{1}^{5} (5 - x)\,dx + \int_{5}^{6} (x - 5)\,dx \]

Step 2: First integral \[ \int_{1}^{5} (5 - x)\,dx = \left[5x - \frac{x^2}{2}\right]_1^5 \] \[ = \left(25 - \frac{25}{2}\right) - \left(5 - \frac{1}{2}\right) = \frac{25}{2} - \frac{9}{2} = 8 \]

Step 3: Second integral \[ \int_{5}^{6} (x - 5)\,dx = \left[\frac{(x-5)^2}{2}\right]_5^6 = \frac{1}{2} \]

Step 4: Total area \[ \text{Area} = 8 + \frac{1}{2} = \frac{17}{2} \] Final: \[ {\frac{17}{2}} \]