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The image contains a calculus problem that reads:

**"Find the volume V of the solid obtained by rotating the region bounded by the curves y = x^3 and y = 5x about the line x = 3."**

### Solution:

To find the volume \( V \) of the solid formed by rotating the region bounded by \( y = x^3 \) and \( y = 5x \) about the line \( x = 3 \), we can use the method of cylindrical shells.

#### Step 1: Set up the integral for the volume using the cylindrical shells method.

The volume \( V \) using the cylindrical shells method is given by:

\[
V = 2\pi \int_{a}^{b} (r_{\text{shell}})(h_{\text{shell}}) \, dx
\]

Where:
- \( r_{\text{shell}} = \text{distance from the axis of rotation} = 3 - x \)
- \( h_{\text{shell}} = \text{height of the shell} = 5x - x^3 \) (difference between the functions)

#### Step 2: Determine the bounds of integration.

We need to find the intersection points of the curves \( y = x^3 \) and \( y = 5x \):

\[
x^3 = 5x
\]

\[
x(x^2 - 5) = 0
\]

This gives:

\[
x = 0 \quad \text{or} \quad x = \sqrt{5} \quad \text{or} \quad x = -\sqrt{5}
\]

However, since we're dealing with the region bounded by \( x = 0 \) and \( x = \sqrt{5} \), we will integrate from 0 to \( \sqrt{5} \).

#### Step 3: Set up the integral.

\[
V = 2\pi \int_{0}^{\sqrt{5}} (3 - x)(5x - x^3) \, dx
\]

#### Step 4: Expand and integrate.

Expanding the integrand:

\[
(3 - x)(5x - x^3) = 15x - 3x^3 - 5x^2 + x^4
\]

The integral becomes:

\[
V = 2\

Find the volume V of the solid obtained by rotating the region bounded by the curves y = x^3 and y = 5x about the line x = 3.

Learn how to find the volume of a solid obtained by rotating the region bounded by y = x^3 and y = 5x about the line x = 3. This solution uses the cylindrical shells method, covering key calculus concepts like integration and volume of solids. Ideal for students in Grades 11-12 or early college.

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