The volume of the solid generated by the revolution of the cardioid r = a(1 + cos(theta)) about the x-axis can be found using the method of cylindrical shells.

First, let's express the equation of the cardioid in terms of x. We can use the equation r = a(1 + cos(theta)) and substitute x for r and theta for arccos((x/a) - 1):

x = a(1 + cos(arccos((x/a) - 1)))
x = a(1 + ((x/a) - 1))
x = a + x - a
x = x

So, the equation of the cardioid in terms of x is x = a.

Next, let's find the limits of integration. The cardioid intersects the x-axis at x = a, so the lower limit is a. The upper limit can be found by setting the equation of the cardioid equal to zero:

x = a(1 + cos(theta))
0 = a(1 + cos(theta))
1 + cos(theta) = 0
cos(theta) = -1
theta = pi

Therefore, the limits of integration are from a to pi.

Now, let's express the volume of the solid as a sum of cylindrical shells. The volume of each shell is given by the formula:

V = 2πrh * Δx

where r is the radius of the shell, h is the height of the shell, and Δx is the thickness of the shell.

The radius of each shell is equal to x, which is a constant, so r = a.

The height of each shell can be found by solving for y in terms of x:

x = a
y = a(1 + cos(theta))
y = a(1 + cos(arccos((x/a) - 1)))
y = a(1 + ((x/a) - 1))
y = x

Therefore, the height of each shell is equal to x.

The thickness of each shell is given by Δx, which is the change in x between each shell. Since x = a is a constant, Δx = 0.

Plugging these values into the volume formula, we get:

V = 2π(a)(x) * 0

Since the thickness of each shell is zero, the volume of each shell is zero. Therefore, the volume of the solid generated by the revolution of the cardioid r = a(1 + cos(theta)) about the x-axis is zero.