Use the method of [[../Cylindrical Shell Volumes|Cylindrical Shells]] to find the volume of the region bounded by the curves $x+y=3$ and $x=4-(y-1)^2$ about the $x-axis$. $\huge \begin{align} x&= 3-y \\ x &=4-(y-1)^2 \\ \\ r(y) &= y\\ h(y) &= 4-(y-1)^2-3+y\\ &= 1-(y-1)^2 + y \\ &= 1-y^2+2y-1+y \\ &= -y^2 + 3y \\ \\ 3-y &=4-(y-1)^2 \\ (y-1)^2-y &= 1 \\ y^2 -2y+1-y &= 1 \\ y^2-3y&= 0 \\ y &=\set{0, 3} \\ \\ \end{align}$ $\huge \begin{align} V &= 2\pi \int_{0}^{3}y\pa{-y^2+3y}\d y \\ &=2\pi \int_{0}^3\pa{-y^3+3y^2}\d y \\ &=2\pi \end{align} $