#### To determine

**To find:**

The volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

#### Answer

V=117π5unit3

#### Explanation

**1) Concept:**

i. If the cross section is a washer with inner radius rin and outer radius rout, then area of washer is obtained by subtracting the area of the inner disk from the area of the outer disk,

A=π outer radius2 -π inner radius2

ii. The volume of solid revolution about y-axis,

V= ∫abA(y)dy

**2) Given:**

The region bounded by x=y-12, x-y=1 rotated about the x=-1

**3) Calculation:**

Find the volume by using washers.

The graph of the region bounded by the given curves is

From figure, as region rotate about the line x=-1, so strip is perpendicular to y-axis.

A cross section of the solid is washer with outer radius is 1+y--1= y+2 and inner radius is y-12-(-1)

From the graph, the points of intersection are (1, 0) and (4, 3)

So its area is given by,

Ay=πouter radius2-πinner radius2

Ay=2+y2-(y-12-(-1)) 2

V= ∫03Aydy=∫03[π2+y2-(y-12-(-1)) 2]dy

=∫03π[y+22-y2-2y+22] dy

=∫03π(y2+4y+4)-(y4-4y3+8y2-8y+4)] dy

=π∫03(-y4+4y3-7y2+12y)] dy

=π-y55+4y44-7y33+12y2203

=π-355+34-7333+632-0

=π -2435+81-63+54

V=117π5

**Conclusion:**

Volume of solid obtained by rotating the region bounded by the given curves is V=117π5unit3