Find the exact area of the surface obtained by rotating the curve about the x-axis.
y = x3,    0 ≤ x ≤ 2

To find the exact area of the surface obtained by rotating the curve \( y = x^3 \) about the x-axis over the interval \( 0 \leq x \leq 2 \), we can use the formula for the surface area of a solid of revolution around the x-axis:

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

### Step 1: Find \( \frac{dy}{dx} \)
Given \( y = x^3 \), we first compute the derivative:

\[
\frac{dy}{dx} = 3x^2
\]

### Step 2: Set up the surface area integral
Substitute \( y = x^3 \) and \( \frac{dy}{dx} = 3x^2 \) into the formula for surface area:

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

Simplify the expression inside the square root:

\[
1 + (3x^2)^2 = 1 + 9x^4
\]

So the surface area becomes:

\[
A = 2\pi \int_{0}^{2} x^3 \sqrt{1 + 9x^4} \, dx
\]

### Step 3: Solve the integral
This integral is a bit complex and typically requires substitution or numerical methods. For now, we leave it as is since there's no elementary antiderivative for \( \sqrt{1 + 9x^4} \), but the exact form is:

\[
A = 2\pi \int_{0}^{2} x^3 \sqrt{1 + 9x^4} \, dx
\]

Would you like details on how to approximate this, or any further steps?

### Related Questions:
1. How do you apply numerical methods to solve complex integrals?
2. What other methods can be used to compute surface areas of solids of revolution?
3. Can you find surface area by rotating around the y-axis instead?
4. How would the surface area formula change for different functions, e.g., trigonometric ones?
5. What are the geometric implications of rotating higher-degree polynomials?

### Tip:
Always simplify expressions inside an integral as much as possible before solving, whether by analytical or numerical methods.

Find the exact area of the surface obtained by rotating the curve about the x-axis. y = x^3, 0 ≤ x ≤ 2

Explore how to calculate the surface area of the solid formed by rotating the curve y = x^3 about the x-axis over the interval 0 ≤ x ≤ 2. This detailed solution covers key calculus concepts such as integration, surface area formulas, and the application of the Fundamental Theorem of Calculus. Suitable for college-level calculus students.

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