- Use symmetry to evaluate the following integral. \[ \int(2-|x|) d x \]

To evaluate the integral \(\int (2 - |x|) \, dx\), let's first observe the symmetry of the function involved. The absolute value function, \(|x|\), indicates that the function is symmetrical about the y-axis. This allows us to consider only the positive side of the integral and then double the resulting area for the negative side. The integral from \(0\) to \(2\) can thus be evaluated as follows: 1. For \(x \geq 0\), \(|x| = x\), so the function becomes \(2 - x\). 2. Setting up the integral from \(0\) to \(2\): \[ \int_0^2 (2 - x) \, dx \] 3. Evaluating the integral: \[ = \left[ 2x - \frac{x^2}{2} \right]_0^2 = \left(2(2) - \frac{2^2}{2}\right) - \left(2(0) - \frac{0^2}{2}\right) \] \[ = \left(4 - 2\right) - 0 = 2 \] 4. Since the function is symmetrical, the total area over the interval from \(-2\) to \(2\) becomes: \[ \text{Total area} = 2 \times 2 = 4 \] Thus, the value of the integral \(\int_{-2}^2 (2 - |x|) \, dx\) is \(4\).