Math Help - Volume Generated by Rotating a Region

1. Volume Generated by Rotating a Region

I am trying to find the volume generated by rotating the region bounded by $y=4+x-x^2$ and $y=4-x$ about the y-axis

I got an answer of $4\pi/3$
using this: $\pi \int_0^2 (4+x-x^2)-(4-x)$

I feel as if this volume is to small but since is rotated about the y-axis I dont have any other values to add that I can think of to account for the area rotated not touching the y-axis.

2. Since you are sweeping along the x-direction and rotating about the y-axis, you are using the "shell" method, which means your integral should be of the form:

$2\pi \int r(x) \, h(x) \, dx$

where r(x) is the radius and h(x) is the height, both in terms of x.

You have correctly identified the height as $(4+x-x^2)-(4-x)$ but you are missing the radius and the "2" in the front. Your limits of integration are correct, also.