#### To determine

**To find:**

The volume generated by rotating the region bounded by given curves about x=-1 using the cylindrical shells

#### Answer

V=40π3unit3

#### Explanation

**1) Concept:**

i. If x is the radius of the typical shell then the circumference =2πx and the height is y=f(x).

ii. By the shell method, the volume of the solid by rotating the region under the curve y=f(x) about x-axis from a to b is

V= ∫ab2πx f(x)dx

where, 0≤a<b

**2) Given:**

y=4-2x, y=0, x=0, rotate about x=-1

**3) Calculation:**

As the region is bounded by y=4-2x, y=0, x=0, and rotated about x=-1, draw the region using the given curves.

The graph shows the region and the cylindrical shell formed by the rotation about the line x=-1

It has the radius =x--1=x+1

Therefore, the circumference is 2π(x+1) and height is 4-2x.

To find the points of intersection, equate y=4-2x and y=0.

4-2x=0

x=2

So, a=0 and b=2

So, the volume of the given solid is

V= ∫ab2π x+1fxdx

= ∫022π x+14-2xdx

= 4π∫02x+12-xdx

=4π∫02-x2+x+2dx

=4π-x33+x22+2x02

=4π-233+222+22

=4π-83+2+4

=4π103

=40π3

**Conclusion:**

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

V=40π3unit3