I need to rotate that region around the y-axis. I'm assuming this means I will have to use the shell method. Finding the intersection gives me $x=0 \; and \; x = 4$ So is this how I should set up the problem?
$$V= \int_0^4 2\pi(x)(4x-x^2-0)dx$$
Thank you.
hint: Not to reiterate what you've done,... but for this example, we use the Shell method: $V = \displaystyle \int_{0}^4 2\pi\cdot x\cdot (4x-x^2)dx$ . Can you continue ? Note that in the "calculus book", they said $f(x) = 4x-x^2$ was the "height" of the "cylinder" and the base is a circular shell whose area is $2\pi xdx$.