#### To determine

**To find:**

The exact volume of the solid

#### Answer

V=2π(π4+π3-12π2-6π+48)

#### Explanation

**1) Concept:**

Let S be a solid.

By the shell method, the volume of the solid by rotating the region bounded by the curve y=f(x) about the line x=-1 from a to b is

V= ∫ab2π x+1 fxdx

where, 0≤a≤b

**2) Given:**

y=x3sinx, y=0, 0≤x≤π rotating about x=-1

**3) Calculations:**

Given,

y=x3sinx, y=0, 0≤x≤π

First, draw the graph of the given curves.

From the graph of the given curves,

If the vertical strip rotates about the line x=-1 then the volume of the solid rotating the region bounded by the given curves is

V= ∫ab2πx+1f(x)dx

=∫0π2πx+1(x3sinx)dx

Evaluate the integral by using the CAS,

Write the command,

Integrate[Sin[x]*(2π*x3*(x+1)),{x,0,π}]

Then output is,

2π(48-6π-12π2+π3+π4)

Therefore,

V=2π(π4+π3-12π2-6π+48)

**Conclusion:**

The exact volume of the solid is

V=2π(π4+π3-12π2-6π+48)