Question

The value of \( \int_{-1}^1 |x| \, dx \) is:

• A. \( -2 \)
• B. \( -1 \)
• C. \( 1 \)
• D. \( 2 \)

Hint:

For integrals of absolute value functions, always split the integral at points where the function changes definition.

Answer 1

The Correct Option is C

Step 1: Splitting the integral. The absolute value function \( |x| \) is defined as Splitting the integral at \( x = 0 \): \[ \int_{-1}^1 |x| \, dx = \int_{-1}^0 -x \, dx + \int_0^1 x \, dx \] Step 2: Evaluating the integrals. \[ \int_{-1}^0 -x \, dx = \left[ -\frac{x^2}{2} \right]_{-1}^0 = 0 - \left(-\frac{(-1)^2}{2}\right) = \frac{1}{2} \] \[ \int_0^1 x \, dx = \left[ \frac{x^2}{2} \right]_0^1 = \frac{1^2}{2} - 0 = \frac{1}{2} \] Step 3: Summing the results. \[ \int_{-1}^1 |x| \, dx = \frac{1}{2} + \frac{1}{2} = 1 \] Conclusion: Thus, the value of the integral is \( 1 \), which corresponds to option \( \mathbf{(C)} \).