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This image shows a problem involving the surface area of a solid of revolution, which appears to be a function \( y = \sqrt{81 - x^2} \) (the equation of a circle with radius 9) rotated around the x-axis. You're asked to evaluate the integral:

\[
2\pi \int_{-6}^{6} (9) \, dx
\]

Here's a breakdown of the steps to solve this problem:

### Step 1: Understand the Geometry
The function \( y = \sqrt{81 - x^2} \) describes the upper half of a circle centered at the origin with radius 9. We are calculating the surface area when this curve is revolved around the x-axis. This specific setup corresponds to calculating the lateral surface area of a solid of revolution.

### Step 2: Simplify the Integral
The integral you've been asked to compute is:

\[
2\pi \int_{-6}^{6} 9 \, dx
\]

This is a straightforward integral of a constant function, 9.

### Step 3: Evaluate the Integral
We can now integrate:

\[
\int_{-6}^{6} 9 \, dx = 9x \Big|_{-6}^{6} = 9(6 - (-6)) = 9(6 + 6) = 9 \times 12 = 108
\]

### Step 4: Multiply by \( 2\pi \)
Finally, multiply the result of the integral by \( 2\pi \):

\[
2\pi \times 108 = 216\pi
\]

### Final Answer:
The value of the given integral is \( 216\pi \).

If you have any questions or need more details, feel free to ask!

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Here are 5 related questions for further understanding:

1. How does the formula for the surface area of revolution work for different curves?
2. What would change in the integral if the limits of integration were different?
3. How can we extend this method to 3D surfaces?
4. What is the role of symmetry in simplifying this type of integral?
5. How would the setup differ if the curve were rotated around the y-axis?

**Tip:** When evaluating definite integrals, always check for symmetry in the function to potentially simplify the problem.

Evaluate the integral 2π∫(-6 to 6) (9) dx for the function y = sqrt(81 - x^2).

This problem involves calculating the surface area of a solid of revolution. It uses the function y = sqrt(81 - x^2) and requires evaluating the definite integral 2π∫(-6 to 6) (9) dx. Learn the steps, key concepts like circle geometry and integration, and how to apply the surface area of revolution formula.

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