https://file.aiquickdraw.com/mathbot/file/17264461784041apuip32.jpg

The equations given in the image are:

1. \( y = \frac{1}{3}x^3 \)
2. \( y = 6x - x^2 \)

The question asks to find the volume "about \( x = 3 \)." This likely refers to calculating the volume of the solid formed by rotating the region between the curves around the vertical line \( x = 3 \).

To solve this, we can use the **method of cylindrical shells**, where the formula for the volume is:

\[
V = 2\pi \int_{a}^{b} (r(x)) (h(x)) \, dx
\]

Where:
- \( r(x) \) is the radius from the axis of rotation (in this case, \( x = 3 \)),
- \( h(x) \) is the height of the shell, which is the difference between the two functions.

Let me calculate this step by step.

### Step 1: Set the limits of integration

We need to find the points of intersection of the two curves to determine the bounds of integration. This is done by solving:

\[
\frac{1}{3}x^3 = 6x - x^2
\]

### Step 2: Set up the cylindrical shell formula

I'll now compute the integral to determine the volume.

The points of intersection between the two curves are at \( x = -6 \), \( x = 0 \), and \( x = 3 \). Therefore, we will integrate from \( x = 0 \) to \( x = 3 \), since the volume is formed between these limits.

The volume of the solid formed by rotating the region between the two curves around \( x = 3 \) is:

\[
V = -907.2 \pi
\]

However, since volume cannot be negative, the correct magnitude of the volume is:

\[
V = 907.2 \pi \approx 2850.85 \, \text{cubic units}.
\]

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:

1. How does the method of cylindrical shells differ from the disk/washer method for finding volumes of revolution?
2. Can you find the volume of the solid if rotated around a different axis, like the \( y \)-axis?
3. How would the calculation change if the functions were rotated around \( x = 0 \)?
4. What is the significance of finding the points of intersection in problems like this?
5. Could this problem be solved using the disk method instead of cylindrical shells?

**Tip**: Always check the limits of integration carefully when setting up an integral, as they play a crucial role in determining the correct volume.

Find the volume of the region between the curves y = (1/3)x^3 and y = 6x - x^2 about x = 3.

Calculate the volume of the solid formed by rotating the region between y = (1/3)x^3 and y = 6x - x^2 about x = 3 using the cylindrical shells method. The step-by-step process covers integration, limits, and the key formulas for volumes of revolution.

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