#### To determine

**To set up:**

An integral for the volume of a solid.

#### Answer

V= ∫-π3π32ππ2-xcos2x-14dx

#### Explanation

**1) Concept:**

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

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

V= ∫ab2πxf(x)dx;a≤x≤b

where 2πx is circumference, fx is height, and dx is the thickness of the shell.

**2) Given:**

y=cos2x, x≤π2, y= 14;about x=π2

**3) Calculation:**

The region is bounded by

y=cos2x, x≤π2, y= 14; about x=π2

Draw the graph of the given curves.

The graph shows the region and representative strip to rotate about the line

x=π2

It has radius

=π2-x

Using shell method, find the approximating shell with radius

π2-x.

Therefore, circumference is

2ππ2-x

and height is

fx=cos2x-14

y=cos2x & y=14

Intersect when

cos2x=14 ⇔ cosx=±12 ⇔x=±π3±2nπ

But

x≤π2

Hence,

x=±π3

So,

a=-π3 and b=π3

So the volume of the given solid is

V= ∫-π3π32ππ2-xfxdx

V= ∫-π3π32ππ2-xcos2x-14dx

**Conclusion:**

The integral for the volume of a solid obtained by rotating the region bounded by the given curves about the line

x=π2

is

V= ∫-π3π32ππ2-xcos2x-14dx