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-xcos4⁡xdx

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=cos4⁡x, y=-cos4⁡x, -π2≤x≤π2; about x=π

3) Calculation:

As the region is bounded by

y=cos4⁡x, y=-cos4⁡x, -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.

cos4⁡x--cos4⁡x=2cos4⁡x

Therefore, the circumference is 2ππ-x and the height is 2cos4⁡x.

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-x2cos4⁡xdx

=4π ∫-π2π2π-xcos4⁡xdx

=4π ∫-π2π2πcos4⁡xdx-4π ∫-π2π2xcos4⁡xdx

By using the theorem of the symmetric function, since cos4⁡x is even, and xcos4⁡x is odd

=8π2∫0π2cos4⁡xdx

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∫0π2cos4⁡xdx