#### 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=13π3unit3

#### Explanation

**1) Concept:**

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

V= ∫ab2πyf(y)dy

where, 0≤a<b

**2) Given:**

x=2y2, y≥0, x=2; about y=2

**3) Calculation:**

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

2y2=2

y=-1, 1

y=-1 is not in the domain.

So, a=0 and b=1

So, the volume of the given solid is

V= ∫ab2π(2-y)fydy

=∫012π2-y2-2y2dy

=∫014π2-y1-y2dy

=4π∫01y3-2y2-y+2dy

=4πy44-2y33-y22+2y01

=4π144-2133-122+21-0

=4π1312

=13π3

**Conclusion:**

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

V=13π3unit3