#### To determine

**To find:**

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

#### Answer

V=27π2unit3

#### 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) about x- axis from a to b is

V= ∫ab2πy f(y)dy

where, 0≤a≤b

**2) Given:**

The region bounded by x+y=4 , x=y2-4y+4 rotated about the x- axis.

**3) Calculation:**

As the region is bounded by x+y=4 ⇒x=4-y, x=y2-4y+4

Using the shell method, the typical approximating shell with the radius y is

To find the height of the strip, subtract the above functions.

4-y-(y2-4y+4)

=-y2+3y

Therefore, the circumference is 2πy and the height is (-y2+3y)

To find the point of intersections,

4-y=(y2-4y+4)

4-y-(y2-4y+4)=0

-y2+3y=0

y3-y=0

y=0 and y=3

Therefore, a=0 and b=3

So, the total volume is

V= ∫ab2πy [-y2+3y ] dy

V=∫032πy [-y2+3y]dy

V=2π∫03-y3+3y2dy

V=2π-y44+3y3303

=-2π 344-33-0

=-2π814-27

V=-2π-274

V=27π2

**Conclusion:**

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

V=27π2 unit3