$$\mathrm{x = 4y-y^2, x = 0, y = 4…}$$ Volume by shell method.
I am just confused with everything. I cannot figure out, the radius, and height. Any help would be appreciated thanks. The radius I took was (4-y) and the function (right - left) which came $\mathrm{(4y-y^2)}$
$$\mathrm{A = 2 \pi \int [4-y] \left[4y-y^2\right] dy}$$
So, I am unsure if I should subtract $$\mathrm{4y - y^2 \ or \ y^2 - 4y}.$$ There is no clear top or bottom/ left or right function and only one point of intersection. Thanks.
First here is the graph:
Here is a graph of this rotated about y=0 or y=4 shifted down.
The range for y, the integration variable is $0\le y\le 4$: $$\mathrm{A=2\pi\int_0^4(4-y)(4y-y^2)dy=2\pi\int_0^4y(4y-y^2)dy=2\pi\int_0^4 4y^2-y^3dy=2\pi\bigg[\frac{4y^3}{3}-\frac{y^4}{4}\bigg]_0^4=\frac{128}3\pi\ (units)=134.04128655...}$$
Try switching the $4y-y^2$ and you will see that this just makes the answer negative. Try also with the (4-y) term and you will see this has the same value. Please correct me and provide me feedback!