Question

Evaluate the integral: \[ \int_{0}^{3} |x - 2| \,dx \] is equal to

• A. \( \frac{2}{3} \)
• B. \( \frac{3}{2} \)
• C. \( \frac{5}{2} \)
• D. \( \frac{2}{5} \)
• E. \( \frac{9}{2} \)

Hint:

When integrating absolute value functions, always split the integral at points where the function changes sign.

Answer 1

The Correct Option is C

Since the function \( |x - 2| \) changes behavior at \( x = 2 \), we split the integral into two parts: \[ \int_{0}^{3} |x - 2| \,dx = \int_{0}^{2} (2 - x) \,dx + \int_{2}^{3} (x - 2) \,dx \] Evaluating each integral: \[ \int_{0}^{2} (2 - x) \,dx = \left[ 2x - \frac{x^2}{2} \right]_{0}^{2} = \left( 4 - 2 \right) - \left( 0 - 0 \right) = 2 \] \[ \int_{2}^{3} (x - 2) \,dx = \left[ \frac{x^2}{2} - 2x \right]_{2}^{3} = \left( \frac{9}{2} - 6 \right) - \left( \frac{4}{2} - 4 \right) \] \[ = \left( \frac{9}{2} - 6 \right) - \left( 2 - 4 \right) = \left( \frac{9}{2} - \frac{12}{2} \right) + 2 = \left( \frac{-3}{2} + 2 \right) = \frac{1}{2} \] Adding both parts: \[ 2 + \frac{1}{2} = \frac{5}{2} \] Thus, the correct answer is (C) \( \frac{5}{2} \).