Find the surface area of the volume generated when
the curve y = x revolves around the x-axis from (1, 1)
to (4, 2)

To find the surface area of the volume generated when the curve \( y = x \) revolves around the x-axis from \( x = 1 \) to \( x = 4 \), we can use the formula for the surface area of a solid of revolution:

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

### Step 1: Identify the components
- \( y = x \), so \( \frac{dy}{dx} = 1 \)
- \( a = 1 \) and \( b = 4 \) are the limits of integration.

### Step 2: Apply the formula
Substituting \( y = x \) and \( \frac{dy}{dx} = 1 \) into the formula:

\[
A = 2\pi \int_{1}^{4} x \sqrt{1 + (1)^2} \, dx
\]

This simplifies to:

\[
A = 2\pi \int_{1}^{4} x \sqrt{2} \, dx
\]

### Step 3: Perform the integration
We can factor out \( \sqrt{2} \):

\[
A = 2\pi \sqrt{2} \int_{1}^{4} x \, dx
\]

The integral of \( x \) is:

\[
\int x \, dx = \frac{x^2}{2}
\]

Now, evaluate this from 1 to 4:

\[
A = 2\pi \sqrt{2} \left[\frac{4^2}{2} - \frac{1^2}{2}\right]
\]
\[
A = 2\pi \sqrt{2} \left[\frac{16}{2} - \frac{1}{2}\right]
\]
\[
A = 2\pi \sqrt{2} \left[\frac{15}{2}\right]
\]
\[
A = 15\pi \sqrt{2}
\]

### Final Answer:
The surface area is \( 15\pi \sqrt{2} \).

Would you like further details on this solution or any clarifications?

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Here are five related questions to explore:
1. How do you find the volume of revolution for the same curve?
2. What is the surface area for the curve \( y = x^2 \) from \( x = 0 \) to \( x = 1 \)?
3. How does the surface area formula change for rotation around the y-axis?
4. What happens if we revolve the curve \( y = \sin(x) \) around the x-axis?
5. How do you find the arc length of the curve \( y = x \) from \( x = 1 \) to \( x = 4 \)?

**Tip:** When using the surface area formula for solids of revolution, make sure to carefully handle the derivative term \( \frac{dy}{dx} \) to avoid mistakes during simplification.

Find the surface area of the volume generated when the curve y = x revolves around the x-axis from (1, 1) to (4, 2).

Explore how to find the surface area of a solid formed by rotating the curve y = x around the x-axis from x = 1 to x = 4. This step-by-step solution uses calculus principles, including the surface area of revolution formula and definite integration, suitable for Grades 11-12.

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