Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

y = (64 − x^2)^(1/4), y = 0, x = 6, x = 7; about the x-axis

v = ∫[6,7] πr^2 dx

where r = y = ∜(64-x^2)

v = π∫[6,7] √(64-x^2) dx
= π(1/2 √(64-x^2) + 32 arcsin(x/8)) [6,7]
= π(√15/2 - √7 + 32(arcsin(7/8)-arcsin(3/4)))

To find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis, we can use the method of cylindrical shells.

First, let's sketch the given curves and the region bounded by them:

The curve y = (64 - x^2)^(1/4) represents the upper boundary of the region between x = 6 and x = 7. The y-axis and x-axis form the other two boundaries. Here's a rough sketch of the region:
```
|
| /|
| / |
| / |
| (x,y) / |
| |¯¯¯¯¯¯¯¯¯¯|
-----+--------+--------+--------+-------+
| x=6 x=7
```
To apply the method of cylindrical shells, we'll consider thin cylindrical shells within this region. Each shell will have a radius (r) equal to the value of x, and the height (h) of each shell will be the difference between the upper and lower boundaries at that particular x-value.

The volume of each cylindrical shell is given by the formula:
V_shell = 2πrh

To calculate the total volume, we need to integrate the volume of each shell over the interval [6, 7] (the x-values where the region is bounded). This can be expressed as follows:
V = ∫(6,7) 2πrh dx

Now, we need to find the expressions for r and h in terms of x.

r (radius) is simply equal to x.

h (height) is the difference between the upper boundary given by y = (64 - x^2)^(1/4), and the lower boundary y = 0:
h = (64 - x^2)^(1/4) - 0
= (64 - x^2)^(1/4)

Therefore, the expression for the volume V becomes:
V = ∫(6,7) 2πx(64 - x^2)^(1/4) dx

However, this integral is not easy to evaluate directly, so we can use a numerical integration method or software to find the value of the integral.

Hope this explanation helps in understanding the approach to finding the volume of the solid obtained by rotating the region bounded by the curves about the x-axis.