#### To determine

**To find:**

The volume of the solid generated by rotating the region bounded by the given curves about specific axis by any method.

#### Answer

V=128π3unit3

#### Explanation

**1) Concept:**

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

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

V= ∫ab2π(y-l)g(y)dy

where 0≤a≤b and g(y)=x

**2) Given:**

The region bounded by x=y-32 , x=4 and rotated about the line y=1

**3) Calculation:**

As region is bounded by x=y-32, x=4 and rotated about the line y=1

Using the shell method,

Radius is (y-1) the circumference is 2π(y-1) and the height is 4-y-32

From the graph, y=1 and y=5 are the y coordinates of the intersection of these two curves.

Therefore, a=1 and b=5

So, the total volume is

V= ∫ab2π(y-1)[4-y-32]dy

V=2π ∫15(-y3+7y2-11y+5 )dy

V=2π-y44+7y33-11y22+5y15

V=2π-544+7533-11522+5(5)--(1)44+7(1)33-11122+5(1)

V=2π27512-1912

V=128π3

**Conclusion:**

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

V=128π3unit3