Question

use the method of cylindrical shells to find the volume generated by rotating the region bounded by .... x=y^2 x=2 about y=-2

Answer 1

the general shell method for rotating around horizontal axis is:
\[V = 2 \pi \int\limits_{a}^{b}r(y) h(y) dy\]
where r is radius and h is height
from graph,
r = 2+y
h = 2-y^2
the bounds are where x=y^2 = 2 --> y = +-sqrt2
\[V = 2 \pi \int\limits_{-\sqrt{2}}^{\sqrt{2}}(2+y)(2-y^{2}) dy\]