#### 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= 8π ∫0π2π-xcos4xdx

#### 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

iii. The theorem of the symmetric function:

Suppose f is continuous on -a, a,

a) If f is an even function, then

∫-aafxdx=2∫0afxdx

**b) ** If f is an odd function, then

∫-aaf(x)dx=0

**2) Given:**

y=cos4x, y=-cos4x, -π2≤x≤π2; about x=π

**3) Calculation:**

As the region is bounded by

y=cos4x, y=-cos4x, -π2≤x≤π2; about x=π

Draw the region using the given curves.

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

It has the radius =π-x

Now, find the height by subtracting the functions.

cos4x--cos4x=2cos4x

Therefore, the circumference is 2ππ-x and the height is 2cos4x.

It is given that x=-π2, π2

So, a=-π2 and b=π2

So, the volume of the given solid is

V= ∫-π2π22π π-xfxdx

= ∫-π2π22π π-x2cos4xdx

=4π ∫-π2π2π-xcos4xdx

=4π ∫-π2π2πcos4xdx-4π ∫-π2π2xcos4xdx

By using the theorem of the symmetric function, since cos4x is even, and xcos4x is odd

=8π2∫0π2cos4xdx

**Conclusion:**

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

V= 8π2∫0π2cos4xdx

**b)** **To evaluate:**

The integral ofthe volume of the solid.

**Solution:**

≈46.50942

**1) Concept:**

Use calculator of integration to evaluate integral

**2) Calculation:**

We have,

V= 8π2∫0π2cos4xdx

By using calculator of integration,

≈46.50942

**Conclusion:**

Therefore,

8π ∫0π2π-xcos4x≈46.50942