#### To determine

**To find:**

The exact volume of the solid

#### Answer

π332

#### Explanation

**1) Concept:**

Let S be the 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=π2 from a to b is

V= ∫ab2π π2-xfxdx

where 0≤a≤b

**2) Given:**

y=sin2x, y=sin4x 0≤x≤π

**3) Calculations:**

Given,

y=sin2x, y=sin4x 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=π2, then the volume of the solid rotating the region bounded by the given curves is

V= ∫ab2ππ2-xf(x)dx

=∫0π/22ππ2-xsin2x-sin4xdx

That is,

=∫0π/22ππ2-xsin2xdx-∫0π/22ππ2-xsin4xdx

Evaluate the integral by using the CAS,

For first integration, write the command,

Integrate[2π(π2-x)(Sin[x])^2,{x,0,π/2}]

Therefore, output is,

18π(-4+π2), and

For Second integration,

Write the command,

Integrate[2π(π2-x)(Sin[x])^4,{x,0,π/2}]

So, output is,

132π(-16+3π2)

To find volume (to subtract this two integrations) write the command,

Subtract[Out[11],Out[12]]

So, output is,

18π(-4+π2)-132π(-16+3π2)

Simplify this by using the command,

Simplify[Out[21]]

So, the output is,

π332

Therefore, the volume of solid is,

V=π332

**Conclusion:**

The exact volume of the solid is

V=π332