Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve about the x-axis.
y =
1
/3
x^3 from 0 to 3

The problem asks for the area of the surface generated by revolving the curve \( y = \frac{1}{3}x^3 \) from \( x = 0 \) to \( x = 3 \) around the x-axis. To calculate the surface area of revolution around the x-axis, we use the formula for surface area \( S \) of a curve \( y = f(x) \) rotated around the x-axis:

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

### Step 1: Define the function and its derivative
Given \( y = \frac{1}{3}x^3 \), let's first find the derivative of \( y \) with respect to \( x \).

\[
\frac{dy}{dx} = x^2
\]

### Step 2: Set up the integral
Now substitute \( y = \frac{1}{3}x^3 \) and \( \frac{dy}{dx} = x^2 \) into the formula:

\[
S = 2\pi \int_{0}^{3} \left( \frac{1}{3}x^3 \right) \sqrt{1 + (x^2)^2} \, dx
\]

This simplifies to:

\[
S = 2\pi \int_{0}^{3} \frac{1}{3}x^3 \sqrt{1 + x^4} \, dx
\]

### Step 3: Evaluate the integral
This integral does not have an elementary antiderivative, so we will approximate the value numerically. Let's calculate it.The surface area generated by revolving the curve \( y = \frac{1}{3}x^3 \) from \( x = 0 \) to \( x = 3 \) around the x-axis is approximately \( 258.85 \) square units.

Would you like further details or have any questions?

Here are 5 related questions to expand on this concept:
1. How would the surface area change if we rotated around the y-axis instead?
2. Can we compute the surface area for a different range, say from \( x = 1 \) to \( x = 3 \)?
3. What would happen to the surface area if the function were \( y = \frac{1}{2}x^2 \)?
4. How can we find the volume generated by rotating this curve around the x-axis?
5. How does the surface area formula differ if we rotate around a line other than the x-axis?

**Tip**: When setting up surface integrals, carefully handle the derivative term, as it significantly influences the final result!

Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve y = 1/3 x^3 from 0 to 3 about the x-axis.

Calculate the surface area of the curve y = 1/3 x^3 rotated around the x-axis from x = 0 to x = 3 using the surface area of revolution formula. This step-by-step solution covers key calculus concepts such as the definite integral and surface area of revolution. Suitable for college-level calculus students.

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