2. Use the washer method to find the volume of the solid of revolution generated by rotating the region bounded by \( y=6-x^{2} \) and \( y=2 \) about the \( x \)-axis.

To find the volume using the washer method, we first need to determine the points of intersection between the curves \( y = 6 - x^2 \) and \( y = 2 \). Setting them equal gives \( 6 - x^2 = 2 \), leading to \( x^2 = 4 \) or \( x = \pm 2 \). For the integration, the volume \( V \) is given by the formula: \[ V = \pi \int_{-2}^{2} \left( (6 - x^2)^2 - 2^2 \right) \, dx. \] Calculating the integral yields: \[ V = \pi \int_{-2}^{2} \left( (36 - 12x^2 + x^4) - 4 \right) \, dx = \pi \int_{-2}^{2} (32 - 12x^2 + x^4) \, dx. \] This results in \( V = \pi \left[ 32x - 4x^3 + \frac{x^5}{5} \right]_{-2}^{2} \). Plugging in the bounds gives the total volume after evaluating. To avoid mistakes, remember to clearly define the outer and inner radii when using the washer method. The outer radius here comes from \( y = 6 - x^2 \) and the inner from \( y = 2\). Ensure to square both functions carefully before performing the integration, and double-check your limits of integration, which are the x-values found from setting the curves equal. A slip in any of these steps can lead to an erroneous volume calculation!