The volume generated by rotating the region bounded by the given curves about y=2 using the cylindrical shells.
i. If y is the radius of the typical shell, then the circumference =2πy and the height is y=f(x)
ii. By the shell method, the volume of the solid by rotating the region under the curve x=f(y) from a to b, about x -axis is
x=2y2, y≥0, x=2; about y=2
As the region is bounded by x=2y2, y≥0, x=2, and rotated about y=2, draw the region using the given curves.
The graph shows the region and the height of typical cylindrical shell formed by the rotation about the line y=2
It has the radius =2-y
Therefore, the circumference is 2π(2-y) and the height is x=2-2y2
To find the points of intersection, equate x=2 and x=2y2. Thus
y=-1 is not in the domain.
So, a=0 and b=1
So, the volume of the given solid is
The volume of solid obtained by rotating the region bounded by the given curves is