The integral represents the volume of a solid of revolution.
2𝜋
4    0 
x^3 dx 

 
(a) Identify the plane region that is revolved.
plane region bounded by y = x2,     y = 0,     x = 0,     x = 4
plane region bounded by y = x1/2,     y = 0,     x = 0,     x = 4    
plane region bounded by y = x2,     y = 16,     y = 0,     x = 0
plane region bounded by x = y2,     x = 0,     y = 0,     y = 4
plane region bounded by y = x3,     y = 0,     x = 0,     x = 4
plane region bounded by y = x3,     y = 16,     y = 0,     x = 0
(b) Identify the axis of revolution.
revolved around the line y = 4
revolved around the line y = 16    
revolved about the y-axis
revolved around the line x = 4
revolved around the line x = 16
revolved about the x-axis

Let's break down the problem and answer each part.

### Part (a): Identifying the plane region that is revolved

The given integral is:

\[
2\pi \int_0^4 x^3 \, dx
\]

This represents the volume of a solid of revolution. When dealing with solids of revolution, the volume is typically given by integrating a function that represents the area of cross-sections formed by revolving a region around an axis.

The term \(x^3\) in the integral suggests that the region bounded by \(y = x^3\), \(y = 0\), and the limits \(x = 0\) to \(x = 4\) is being revolved. These boundaries form a typical region under the curve.

Thus, the correct plane region is:
- **Plane region bounded by \( y = x^3 \), \( y = 0 \), \( x = 0 \), \( x = 4 \)**

### Part (b): Identifying the axis of revolution

The integral also contains a factor of \(2\pi\), which is a strong indication that the region is being revolved around the **x-axis**, as the formula for the volume of a solid of revolution using the disk method is \( V = 2\pi \int x f(x) \, dx \), where \(f(x)\) is the function being revolved.

Therefore, the axis of revolution is:
- **Revolved about the x-axis**

### Final Answer:
- (a) **Plane region bounded by \( y = x^3 \), \( y = 0 \), \( x = 0 \), \( x = 4 \)**
- (b) **Revolved about the x-axis**

Would you like further details or explanations?

Here are five follow-up questions for better understanding:
1. How does the disk method work for finding the volume of a solid of revolution?
2. What happens to the volume if the axis of revolution changes?
3. How would you compute the volume if the plane region was bounded by \(y = x^2\) instead of \(y = x^3\)?
4. Can you describe the solid formed by revolving \(y = x^3\) around the x-axis visually?
5. How does the factor of \(2\pi\) relate to the geometry of revolution?

**Tip:** Always check the structure of the integral (such as the presence of \(2\pi\)) to quickly identify the method being used (disk/washer or shell) in a volume problem.

The integral represents the volume of a solid of revolution.
2π ∫ from 0 to 4 of x^3 dx 

(a) Identify the plane region that is revolved.
(b) Identify the axis of revolution.

Learn how to calculate the volume of a solid of revolution by solving the integral of x^3 from 0 to 4, revolved around the x-axis. This solution uses the disk method and explores key calculus concepts like solids of revolution and integration. Suitable for college calculus students or advanced high school learners.

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