#### 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=4π3unit3

#### Explanation

**1) Concept:**

i. If the cross section is the disc and the radius of the disc is in terms of x or y, then

A=π radius2

ii. The volume of the solid revolution about the y-axis is

V= ∫abA(y)dy

**2) Given:**

The region bounded by x2+y-12=1, rotated about the y- axis.

**3) Calculation:**

Find the volume by using the disks.

The graph of the region bounded by the given curves is

x2+y-12=1

Solve for x from this

x=±1-y-12

A cross section of the solid is the disk with the radius 1-y-12

So, its area is given by

Ay=π1-y-122

V= ∫02Aydy=∫02π1-y-122dy

=∫02π1-y-12dy

=∫02π1-y2+2y-1dy

=∫02π-y2+2ydy

=π-y33+2y2202

=π -(2)33+2(2)22-0

=π-83+4

V=4π3

**Conclusion:**

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

V=4π3unit3