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To find the exact surface area generated by rotating the curve \( y = \sqrt{x^2 + 1} \) from \( x = 0 \) to \( x = 6 \) about the x-axis, we can use the formula for the surface area of revolution:

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

### Step 1: Differentiate \( y = \sqrt{x^2 + 1} \)
We first need to find the derivative of \( y = \sqrt{x^2 + 1} \).

\[
\frac{dy}{dx} = \frac{d}{dx} \left( \sqrt{x^2 + 1} \right) = \frac{2x}{2\sqrt{x^2 + 1}} = \frac{x}{\sqrt{x^2 + 1}}
\]

### Step 2: Set up the integral
Now, substitute \( y = \sqrt{x^2 + 1} \) and \( \frac{dy}{dx} = \frac{x}{\sqrt{x^2 + 1}} \) into the formula for the surface area.

\[
S = 2\pi \int_{0}^{6} \sqrt{x^2 + 1} \sqrt{1 + \left( \frac{x}{\sqrt{x^2 + 1}} \right)^2} \, dx
\]

Simplifying the expression under the square root:

\[
1 + \left( \frac{x}{\sqrt{x^2 + 1}} \right)^2 = 1 + \frac{x^2}{x^2 + 1} = \frac{x^2 + 1 + x^2}{x^2 + 1} = \frac{2x^2 + 1}{x^2 + 1}
\]

Thus, the integral becomes:

\[
S = 2\pi \int_{0}^{6} \sqrt{x^2 + 1} \cdot \sqrt{\frac{2x^2 + 1}{x^2 + 1}} \, dx = 2\pi \int_{0}^{6} \sqrt{2x^2 + 1} \, dx
\]

### Step 3: Solve the integral
Now we need to evaluate the integral \( \int \sqrt{2x^2 + 1} \, dx \). This integral can be computed using a substitution method or a table of integrals. Let’s solve this step:

Let \( u = 2x^2 + 1 \), then \( du = 4x dx \), leading to:

\[
S = 2\pi \cdot \left[ \frac{1}{6} (2x^2 + 1)^{3/2} \right]_{0}^{6}
\]

Substituting the limits:

\[
S = 2\pi \left[ \frac{1}{6} \left( 2(6^2) + 1 \right)^{3/2} - \frac{1}{6} (2(0^2) + 1)^{3/2} \right]
\]

Simplifying:

\[
S = 2\pi \left[ \frac{1}{6} \left( 73^{3/2} - 1^{3/2} \right) \right]
\]

### Final answer:
The exact surface area is:

\[
S = \frac{\pi}{3} \left( 73^{3/2} - 1 \right)
\]

Would you like more details on any of the steps? Here are some related questions:

1. How do you find the surface area when rotating around the y-axis?
2. Can you use numerical methods to approximate this integral?
3. How does this method change if the function is rotated about a different axis?
4. What other applications are there for surface area of revolution in physics or engineering?
5. How do different functions affect the complexity of surface area integrals?

**Tip**: Always double-check the function's derivative carefully before applying it in surface area calculations.

Find the exact area of the surface obtained by rotating the given curve about the x-axis. y = sqrt(x^2 + 1), 0 <= x <= 6

Learn how to find the exact surface area obtained by rotating the curve y = sqrt(x^2 + 1) from x = 0 to x = 6 about the x-axis. This problem involves the use of calculus concepts, including differentiation and integration, with a focus on the surface area of revolution.

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