Find the surface area of the surface obtained by rotating the portion of the
curve y = 4 sin(6x) between the points
P = (0, 0) and Q = (
(%, 0) around the
x-Axis.
Use a calculator to evaluate any integral(s), and round your answer to three decimal places.

To find the surface area of the surface obtained by rotating the curve \( y = 4 \sin(6x) \) around the x-axis between the points \( P = (0, 0) \) and \( Q = \left( \frac{\pi}{6}, 0 \right) \), we can use the surface area formula for a curve rotated about the x-axis:

\[
A = 2\pi \int_{a}^{b} y \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx
\]

### Step 1: Set up the problem
Given:
- The curve is \( y = 4 \sin(6x) \),
- The limits of integration are from \( a = 0 \) to \( b = \frac{\pi}{6} \).

### Step 2: Compute the derivative of \( y \)
We need to find \( \frac{dy}{dx} \). For \( y = 4 \sin(6x) \):

\[
\frac{dy}{dx} = 4 \cdot 6 \cos(6x) = 24 \cos(6x)
\]

### Step 3: Set up the surface area integral
Now we can substitute \( y = 4 \sin(6x) \) and \( \frac{dy}{dx} = 24 \cos(6x) \) into the formula for the surface area:

\[
A = 2\pi \int_{0}^{\frac{\pi}{6}} 4 \sin(6x) \sqrt{1 + (24 \cos(6x))^2} \, dx
\]

This simplifies to:

\[
A = 8\pi \int_{0}^{\frac{\pi}{6}} \sin(6x) \sqrt{1 + 576 \cos^2(6x)} \, dx
\]

### Step 4: Evaluate the integral
This integral is quite complex, so we will use a calculator or numerical integration method to compute it. Let me calculate the value for you.

The surface area of the surface obtained by rotating the curve \( y = 4 \sin(6x) \) around the x-axis between the points \( P = (0, 0) \) and \( Q = \left( \frac{\pi}{6}, 0 \right) \) is approximately \( 101.294 \, \text{square units} \), rounded to three decimal places.

Would you like more details on the steps? Here are some related questions to explore:

1. How is the surface area formula for rotation around the x-axis derived?
2. What is the geometrical interpretation of this surface?
3. Can we solve this integral using substitution methods?
4. How would the surface area change if we rotated around the y-axis instead?
5. What happens if we rotate a different trigonometric function, like \( y = \cos(x) \)?

**Tip:** When dealing with complex integrals involving trigonometric functions, numerical methods can often give an accurate solution without extensive algebra.

Find the surface area of the surface obtained by rotating the portion of the curve y = 4 sin(6x) between the points P = (0, 0) and Q = (π/6, 0) around the x-Axis. Use a calculator to evaluate any integral(s), and round your answer to three decimal places.

Find the surface area of the curve y = 4 sin(6x) rotated around the x-axis between P(0,0) and Q(π/6,0). Learn how to apply the surface area formula for a curve and use numerical methods to evaluate complex integrals. Suitable for high school and college-level calculus students.

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