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

To evaluate the integral \[ \int(2 - |x|) \, dx \] using symmetry, we first need to understand the function \( f(x) = 2 - |x| \). ### Step 1: Analyze the function The function \( |x| \) is symmetric about the y-axis, meaning it is an even function. Therefore, \( f(x) \) can be analyzed in two parts: - For \( x \geq 0 \): \( f(x) = 2 - x \) - For \( x < 0 \): \( f(x) = 2 + x \) ### Step 2: Set up the integral To evaluate the integral over a symmetric interval, we can compute it from \( 0 \) to \( a \) and then double the result: \[ \int_{-a}^{a} (2 - |x|) \, dx = 2 \int_{0}^{a} (2 - x) \, dx \] ### Step 3: Calculate the integral from \( 0 \) to \( a \) Now, we compute the integral: \[ \int_{0}^{a} (2 - x) \, dx \] This can be calculated as follows: \[ \int (2 - x) \, dx = 2x - \frac{x^2}{2} + C \] Evaluating from \( 0 \) to \( a \): \[ \left[ 2x - \frac{x^2}{2} \right]_{0}^{a} = \left( 2a - \frac{a^2}{2} \right) - \left( 2(0) - \frac{0^2}{2} \right) = 2a - \frac{a^2}{2} \] ### Step 4: Double the result Now, we double the result to account for the negative side: \[ \int_{-a}^{a} (2 - |x|) \, dx = 2 \left( 2a - \frac{a^2}{2} \right) = 4a - a^2 \] ### Conclusion The integral evaluates to: \[ \int_{-a}^{a} (2 - |x|) \, dx = 4a - a^2 \] If we want to evaluate it over a specific interval, we can substitute \( a \) with the desired limits. For example, if we want to evaluate from \( -2 \) to \( 2 \): \[ \int_{-2}^{2} (2 - |x|) \, dx = 4(2) - (2^2) = 8 - 4 = 4 \] Thus, the final answer for the integral over the interval from \( -2 \) to \( 2 \) is: \[ \int_{-2}^{2} (2 - |x|) \, dx = 4 \]