Use the method of [[../Cylindrical Shell Volumes|Cylindrical Shells]] to find the volume of the region bounded by the curves $y = 3+2x-x^{2}$ and $y+x=3$ about the $y-axis$. ![[../../../00 Asset Bank/Pasted image 20241001152650.png|invert_B]] $\huge \begin{align} V &= 2\pi \int_{0}^{3} x\pa{ 3+2x-x^2 -3+x }\d x \\ &= 2\pi \int_{0}^{3} x\pa{ 3x-x^2 }\d x \\ &= 2\pi \int_{0}^{3} \pa{ 3x^2 -x^3 }\d x \\ &= 2\pi \pa{x^3 - \frac{1}{4}x^4} \big \vert ^3_{0} \\ &=\boxed{2\pi\pa{ 27- \frac{81}{4} }} \end{align} $