#### To determine

**a) **

**To find:**

The integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.

#### Answer

V= 2π∫01x+1x-x41+x3dx

#### Explanation

**1) Concept:**

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

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

V= ∫ab2πx f(x)dx

where 0≤a<b

**2) Given:**

y=x, y=2x1+x3; about x=-1

**3) Calculation:**

As the region is bounded by

y=x, y=2x1+x3; about x=-1

Draw the region using the given curves.

The graph shows the region. A typical cylindrical shell is formed by the rotation about the line x=-1.

It has the radius =x--1=x+1

Therefore, the circumference is

2πx+1 and the height is 2x1+x3-x

Now, to find the intersection points,

2x1+x3=x

x+x4=2x

x4-x=0

xx-1x2+x+1=0

x=0, 1

So, a=0 and b=1

So, the volume of the given solid is

V= ∫012πx+1fxdx

= 2π∫01x+12x1+x3-xdx

= 2π∫01x+1x-x41+x3dx

**Conclusion:**

The integral for the volume of the solid obtained by rotating the region bounded by the given curves about the x=-1 is

V=2π∫01x+1x-x41+x3dx

#### To determine

**b) **

**To evaluate:**

The integral ofthe volume of the solid.

#### Answer

≈2.36164

#### Explanation

**1) Concept:**

Use calculator to evaluate integral

**2) Calculation:**

We have,

V=2π∫01x+1x-x41+x3dx

By using calculator (refer to exercise 21),

≈2.36164

**Conclusion:**

Therefore,

2π∫01x+1x-x41+x3dx≈2.36164