#### 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:**

The volume of the disk obtained by the revolution about the y-axis is

V= ∫abA(y)dy where A(y) is the area of cross-section perpendicular y-axis.

**2) Given:**

The region bounded by y2-x2=1, y=2, rotated about the y- axis.

**3) Calculation:**

The graph of the region bounded by the given curves is

Find the volume by using the area of disks which are perpendicular to y-axis,

Here, the region is bounded by y2-x2=1⇒x=±y2-1, y=2

We are interested only in positive y and the region is bounded in y=[1, 2]. So

V=∫12πy2-12dx

=∫12πy2-1dx

=π13y3-y 12

=π 13(2)3-2 -13(1)3-1

=π 83-2 -13-1

V=π23+23

V=4π3

**Conclusion:**

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

V=4π3unit3