\( =\begin{array}{l}\text { Use symmetry to evaluate the following imtegral, } \\ \int_{-B}^{B}(8-|x|) d x \\ 0\end{array}(8) \)

To evaluate the integral \[ \int_{-B}^{B}(8-|x|) \, dx \] using symmetry, we can follow these steps: 1. **Identify the function**: The integrand is \( f(x) = 8 - |x| \). This function is even because \( f(-x) = 8 - |-x| = 8 - |x| = f(x) \). 2. **Use the property of even functions**: For an even function, the integral from \(-B\) to \(B\) can be simplified as follows: \[ \int_{-B}^{B} f(x) \, dx = 2 \int_{0}^{B} f(x) \, dx \] 3. **Calculate the integral from 0 to \(B\)**: We need to evaluate \[ \int_{0}^{B} (8 - x) \, dx \] 4. **Perform the integration**: \[ \int (8 - x) \, dx = 8x - \frac{x^2}{2} + C \] 5. **Evaluate the definite integral**: \[ \int_{0}^{B} (8 - x) \, dx = \left[ 8x - \frac{x^2}{2} \right]_{0}^{B} = \left( 8B - \frac{B^2}{2} \right) - \left( 8(0) - \frac{0^2}{2} \right) = 8B - \frac{B^2}{2} \] 6. **Multiply by 2 for the full integral**: \[ \int_{-B}^{B} (8 - |x|) \, dx = 2 \left( 8B - \frac{B^2}{2} \right) = 16B - B^2 \] Thus, the final result for the integral is: \[ \int_{-B}^{B} (8 - |x|) \, dx = 16B - B^2 \]