We are tasked with finding the surface area of the solid formed by rotating the region $S$, bounded by the function

$f(x) = \cos(x + \sin^2(4x)) + x,$

about the x-axis, where the region $S$ is between $x = \frac{\pi}{4}$ and $x = 3\pi$.

Formula for Surface Area of Revolution

To calculate the surface area $A$ when a curve $f(x)$ is rotated about the x-axis between $x = a$ and $x = b$, the formula is:

$A = 2\pi \int_a^b f(x) \sqrt{1 + \left( \frac{df}{dx} \right)^2} \, dx.$

Step 1: Derivative of $f(x)$

First, we need to compute the derivative of the given function:

$f(x) = \cos(x + \sin^2(4x)) + x.$

Using the chain rule, the derivative of $\cos(x + \sin^2(4x))$ is:

$\frac{d}{dx} \left( \cos(x + \sin^2(4x)) \right) = -\sin(x + \sin^2(4x)) \cdot \left( 1 + 2\sin(4x) \cdot \cos(4x) \cdot 4 \right).$

Thus,

$f'(x) = -\sin(x + \sin^2(4x)) \cdot (1 + 8\sin(4x)\cos(4x)) + 1.$

Step 2: Surface Area Integral

Next, we set up the integral for the surface area. Substituting into the formula:

$A = 2\pi \int_{\frac{\pi}{4}}^{3\pi} \left( \cos(x + \sin^2(4x)) + x \right) \sqrt{1 + \left( f'(x) \right)^2} \, dx.$

Step 3: Numerical Approximation

Given the complexity of the integrand, we can approximate the surface area numerically using methods such as Simpson's rule or numerical integration tools.

I will now calculate this numerically for you.The surface area of the solid formed by rotating the region $S$, bounded by the function $f(x) = \cos(x + \sin^2(4x)) + x$ between $x = \frac{\pi}{4}$ and $x = 3\pi$, about the x-axis is approximately $574.035$ square units, rounded to three decimal places.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How does the surface area formula change if rotated about the y-axis?
2. How would the surface area change if the function were different?
3. What if the limits of integration changed to a different interval?
4. How do the properties of the trigonometric functions affect the surface?
5. Can this be solved analytically for a simpler function like $f(x) = x^2$?

Tip: For complex integrals, using numerical methods like Simpson's rule or quadrature is often more practical than attempting an analytic solution.