#### To determine

**To find:**

The volume generated by rotating the region bounded by the given curves about y=-2 using the cylindrical shells.

#### Answer

V=16π3unit3

#### Explanation

**1) Concept:**

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

V= ∫ab2πxf(y)dy

where 0≤a<b

iii. The theorem of symmetric function:

Suppose f is continuous on -a, a,

(a) If f is a even function, then

∫-aafxdx=2∫0afxdx

(b) If f is an odd function, then

∫-aaf(x)dx=0

**2) Given:**

x=2y2, x=y2+1; about y=-2

**3) Calculation:**

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

fy=y2+1-2y2=1-y2

To find the point of intersection,

y2+1=2y2

y2=1

y=-1, 1

So, a=-1 and b=1

So, the volume of the given solid is

V= ∫ab2π(2+y)fydy

=∫-112π2+y1-y2dy

=2π∫-112+y1-y2dy

=4π∫01-y3-2y2+y+2dy

By using the theorem of symmetric function, the contribution of y and -y3 to the integral vanishes, so

=4π∫01-2y2+2dy

=8π∫01-y2+1dy

=8π-y33+y01

=8π-133+1-0

=8π23

=16π3

**Conclusion:**

The volume of solid obtained by rotating the region bounded by the given curves is

V=16π3unit3