Find the surface area of revolution about the y-axis of x = sqrt(64 - y^2) over the interval 0 <= y <= 4 

To find the surface area of revolution about the \( y \)-axis for the curve \( x = \sqrt{64 - y^2} \), we use the formula for the surface area of revolution:

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

### Step 1: Set up the equation
We are given:
\[
x = \sqrt{64 - y^2}
\]

First, compute \( \frac{dx}{dy} \).

\[
\frac{dx}{dy} = \frac{d}{dy} \left( \sqrt{64 - y^2} \right) = \frac{-y}{\sqrt{64 - y^2}}
\]

### Step 2: Plug into the surface area formula
Now substitute \( x \) and \( \frac{dx}{dy} \) into the surface area formula. First, calculate \( 1 + \left( \frac{dx}{dy} \right)^2 \):

\[
\left( \frac{dx}{dy} \right)^2 = \frac{y^2}{64 - y^2}
\]

Thus,

\[
1 + \left( \frac{dx}{dy} \right)^2 = 1 + \frac{y^2}{64 - y^2} = \frac{64}{64 - y^2}
\]

Now take the square root:

\[
\sqrt{1 + \left( \frac{dx}{dy} \right)^2} = \frac{8}{\sqrt{64 - y^2}}
\]

### Step 3: Write the surface area integral
Substitute everything into the surface area formula:

\[
S = 2\pi \int_{0}^{4} \sqrt{64 - y^2} \cdot \frac{8}{\sqrt{64 - y^2}} \, dy
\]

The \( \sqrt{64 - y^2} \) terms cancel out:

\[
S = 2\pi \int_{0}^{4} 8 \, dy
\]

### Step 4: Integrate
Now, integrate \( 8 \) with respect to \( y \):

\[
S = 2\pi \left[ 8y \right]_{0}^{4} = 2\pi (8 \times 4 - 8 \times 0) = 64\pi
\]

### Final Answer:
The surface area of revolution is \( 64\pi \) square units.

Would you like more details, or do you have any questions?

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Here are some related questions:
1. How do we find surface area for revolution around the x-axis?
2. Can the formula change if the function is implicit?
3. How do surface area and volume of revolution relate?
4. What if the curve was rotated about a different axis, like \( x = 2 \)?
5. How does changing the interval affect the surface area?

**Tip:** When solving surface area problems, always check for symmetry. This can sometimes simplify calculations significantly!

Find the surface area of revolution about the y-axis of x = sqrt(64 - y^2) over the interval 0 <= y <= 4.

Find the surface area of revolution for the curve x = sqrt(64 - y^2) about the y-axis over the interval 0 ≤ y ≤ 4. This detailed step-by-step solution explores key calculus concepts, including integration and the surface area of revolution formula, suitable for advanced students in Grades 11-12.

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