#### To determine

**To find:**

The volume of a solid obtained by rotating the region bounded by the given curves about the specified line and sketch the region, the solid, and a typical disk or washer.

#### Answer

256π

#### Explanation

**1) Concept:**

i. If the cross section is a washer with the 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 of revolution about x-axis.

V= ∫abA(x)dx

**2) Given:**

The region is bounded by y=x2+1,y=9-x2 rotated about the line y=-1.

**3) Calculation:**

As the region is bounded by y=x2+1,y=9-x2 rotated about the line y=-1,

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

A cross section of the solid is a washer with the outer radius 9-x2--1=10-x2 and inner radius x2+1--1=x2+2.

So its area is given by

Ax=πouter radius2-πinner radius2

=π10-x22-πx2+22

=π(100-20x2+x4)-(x4+4x2+4)

=π100-20x2+x4-x4-4x2-4

=π96-24x2

The region of integration is bounded by y=x2+1,y=9-x2.

Therefore at intersection, x2+1=9-x2, 2x2=8, x2=4

x=-2, x=2

Here the limits of integration are x=-2 to x= 2

Therefore, the volume of the solid revolution about line y=-1,

V= ∫-22Axdx=∫-22π(96-24x2)dx

The graph is symmetric.

Therefore,V=2π∫02(96-24x2)dx

By using fundamental theorem of calculus and power rule of integration,

V=2π96x-24x3302

=2π96x-8x302

By substituting limits of integration,

=2π 96(2)-8(2)3-96(0)-8(0)

=2π192-64=2π(128)

V=256π

**Conclusion:**

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

V=256π