Question

Evaluate the integral: $ \int_{-1}^1 |x - 3| \, dx $

• A. \( 0 \)
• B. \( 3 \)
• C. \( 4 \)
• D. \( 6 \)

Hint:

For integrals involving absolute values, consider breaking the integrand into pieces based on where the function changes its sign, and then compute the integrals separately for each piece.

Answer 1

The Correct Option is D

We are given the integral: \[ \int_{-1}^1 |x - 3| \, dx \] Since \( |x - 3| \) represents the absolute value, we split the integral at the point where \( x = 3 \). In this case, the function \( |x - 3| \) will behave differently depending on whether \( x<3 \) or \( x \geq 3 \). However, since we are integrating from \( -1 \) to \( 1 \), the expression \( |x - 3| \) is always positive, and the integral can be simplified: \[ \int_{-1}^1 (x - 3) \, dx \] Evaluating the integral: \[ = \left[\frac{x^2}{2} - 3x\right]_{-1}^1 = \left(\frac{1^2}{2} - 3(1)\right) - \left(\frac{(-1)^2}{2} - 3(-1)\right) \] \[ = \left(\frac{1}{2} - 3\right) - \left(\frac{1}{2} + 3\right) \] \[ = \left(\frac{-5}{2}\right) - \left(\frac{7}{2}\right) = -6 \] 
Thus, the final answer is \( -6 \).