Find area of solid using cylindrical shell method Here's my problem, I have to use the shell method to find the area bounded between f(x)=4x-x^2 and y=3 by rotating about y=2. I understand I have to first solve for x. Which is weird because I have two x variables with different powers. So does this mean I am going to have x on both sides? How would I use the shell method if this is the case?

You only need one of those equations since both regions we want to rotate are equal regions if that makes sense. So you should solve that equation above and get $x =\pm \sqrt{4-y}+2$. Well use the right hand side of the parabola in rotate it. We will also use the axis of symmetry of our parabola to help guide us to what we need to put in our formula. $2 \cdot \int_3^4 2 \pi (\sqrt{4-y}+2-2) \cdot (y-2) dy$. The factor of 2 in front of the integral there comes from that we need to rotate both sides of $x=2$ about $y=2$. This answer should be equivalent to what we get doing the washer method $\pi \int_1^3 ((4x-x^2-2)^2-(3-2)^2) dx$
