Using the shell method, the volume of the solid generated by revolving the shaded region about the y-axis, given by
. Thus, the correct answer is A.
To find the volume of the solid generated by revolving the shaded region about the y-axis using the shell method, we'll integrate along the y-axis.
The equation
can be rewritten as
. To find the limits of integration, we need to determine the points where the curves intersect:
Factoring out x, we get
. So,
are the points of intersection.
Now, the radius of the shell is the distance from the y-axis to the curve, which is x. The height of the shell is the differential change in y, denoted as dy.
The volume element of a shell is
.
Now, integrate
from
(the square of
):
Evaluate this integral to find the volume.
Therefore, the correct answer is A.
.