$12)$ Consider the region in the first quadrant bounded by $\backslash ($ y$=-$x$^\{2\}+2\backslash ),$ the $\backslash ($ x $\backslash )-$axis, and the $\backslash ($ y $\backslash )-$axis.

a$.$ Sketch over the given graph a picture of the solid that is created when the region is revolved about the vertical line $\backslash ($ x$=\backslash$sqrt$\{2\}\backslash ).$ Notice that is an $\backslash ($ x $\backslash )-$intercept in the picture.

b$.$ Decide whether it is appropriate to take vertical or horizontal slices to find the volume of this solid. Draw one representative slice and be sure to label the thickness of the slice with the appropriate notation of either $\backslash (\backslash$Delta x $\backslash )$ or $\backslash (\backslash$Delta $\backslash$mathrm$\{$y$\}\backslash ).$

c$.$ Give an expression for the volume of this one representative slice. Do not pick specific values of $\backslash ($ x $\backslash )$ or $\backslash ($ y $\backslash )$ to work with. We are talking about a representative slice at location $\backslash ($ x$\_\{$i$\}\backslash )$ or $\backslash ($ y$\_\{$i$\}\backslash ).$