Question

Let f(x)=2√4−x2 over the interval [−1,0]. Use a calculator to find the surface area of the surface generated by revolving the graph of f(x) around the x-axis.

Round your answer to four decimal places.

Provide your answer below:

Surface Area $\approx$≈​  $\text{ units}^2$ units2​




To find the surface area of the surface generated by revolving the graph of $ f(x) = 2\sqrt{4 - x^2} $ around the x-axis over the interval $[-1, 0]$, we will use the surface area formula for revolution around the x-axis:

\[
S = 2\pi \int_{a}^{b} f(x) \sqrt{1 + (f'(x))^2} \, dx
\]

### Step-by-step procedure:

1. **Find the derivative of $ f(x) = 2\sqrt{4 - x^2} $**:
   \[
   f'(x) = \frac{d}{dx}\left( 2\sqrt{4 - x^2} \right)
   \]
   Use the chain rule:
   \[
   f'(x) = 2 \cdot \frac{1}{2}(4 - x^2)^{-1/2} \cdot (-2x)
   \]
   Simplifying:
   \[
   f'(x) = \frac{-2x}{\sqrt{4 - x^2}}
   \]

2. **Substitute $ f(x) $ and $ f'(x) $ into the surface area formula**:
   \[
   S = 2\pi \int_{-1}^{0} 2\sqrt{4 - x^2} \sqrt{1 + \left(\frac{-2x}{\sqrt{4 - x^2}}\right)^2} \, dx
   \]
   Simplify the expression inside the square root:
   \[
   \sqrt{1 + \left(\frac{-2x}{\sqrt{4 - x^2}}\right)^2} = \sqrt{1 + \frac{4x^2}{4 - x^2}} = \sqrt{\frac{4 - x^2 + 4x^2}{4 - x^2}} = \sqrt{\frac{4 + 3x^2}{4 - x^2}}
   \]

3. **Set up the final integral**:
   \[
   S = 4\pi \int_{-1}^{0} \sqrt{4 - x^2} \cdot \sqrt{\frac{4 + 3x^2}{4 - x^2}} \, dx
   \]
   This integral is best solved numerically.

### Numerical Calculation:

Let me compute this integral for you.The surface area of the surface generated by revolving the graph of $ f(x) = 2\sqrt{4 - x^2} $ around the x-axis over the interval $[-1, 0]$ is approximately:

\[
\text{Surface Area} \approx 55.9824 \, \text{units}^2
\]

Would you like more details on the steps, or do you have any further questions?

Here are 5 related questions to consider:
1. How does the surface area formula change for revolutions around the y-axis?
2. What if the function $ f(x) $ was revolved over a different interval, say $[0, 1]$?
3. How would the surface area change if $ f(x) $ was defined as $ \sqrt{4 - x^2} $ instead of $ 2\sqrt{4 - x^2} $?
4. Can you find the surface area for revolution of other functions like $ f(x) = e^x $?
5. How is the derivative $ f'(x) $ used in surface area calculations?

**Tip**: When solving problems involving surfaces of revolution, always carefully handle derivatives and integrals, as they can drastically affect the result.

Let f(x) = 2√4 − x^2 over the interval [−1, 0]. Use a calculator to find the surface area of the surface generated by revolving the graph of f(x) around the x-axis. Round your answer to four decimal places.

Calculate the surface area of the surface generated by revolving the graph of f(x) = 2√4 − x² around the x-axis, over the interval [−1, 0]. Using calculus techniques like definite integration and the chain rule, the solution provides a step-by-step guide for advanced students. The final surface area is approximately 55.9824 square units.

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