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 x=f(y)
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
iii. The theorem of symmetric function:
Suppose f is continuous on -a, a,
(a) If f is a even function, then
(b) If f is an odd function, then
x=2y2, x=y2+1; about y=-2
As the region is bounded by x=2y2, x=y2+1; about y=-2, draw the graph of 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
Using the shell method, to find the typical approximating shell with the radius 2+y.
Therefore, the circumference is 2π(2+y) and the height is
To find the point of intersection,
So, a=-1 and b=1
So, the volume of the given solid is
By using the theorem of symmetric function, the contribution of y and -y3 to the integral vanishes, so
The volume of solid obtained by rotating the region bounded by the given curves is