y=x^3/2, y=8,x=0 about the x axis by cylindrical shells

We are tasked with finding the volume of a solid generated by rotating the curve \( y = \frac{x^3}{2} \) around the x-axis using the method of cylindrical shells. The given information is \( y = 8 \) and \( x = 0 \).

### Step 1: Problem Setup

The volume of a solid of revolution using cylindrical shells is given by the formula:

\[
V = \int_{a}^{b} 2\pi \cdot \text{radius} \cdot \text{height} \, dx
\]

- **Radius**: In this case, the radius is the distance from the x-axis, which is simply \( x \).
- **Height**: The height of each shell is the value of \( y \), which is given by the function \( y = \frac{x^3}{2} \).

We are rotating about the x-axis, and we are given that \( y = 8 \), so we need to find the limits of integration for \( x \).

### Step 2: Finding the limits of integration

We are given that \( y = 8 \), so we set the function \( y = \frac{x^3}{2} \) equal to 8 to find the upper limit of integration:

\[
\frac{x^3}{2} = 8
\]
\[
x^3 = 16
\]
\[
x = \sqrt[3]{16}
\]

So, the upper limit of integration is \( x = \sqrt[3]{16} \). The lower limit is \( x = 0 \), as stated in the problem.

### Step 3: Set up the integral

Now we can set up the integral for the volume using the method of cylindrical shells:

\[
V = \int_{0}^{\sqrt[3]{16}} 2\pi \cdot x \cdot \frac{x^3}{2} \, dx
\]

Simplifying the integrand:

\[
V = 2\pi \int_{0}^{\sqrt[3]{16}} x \cdot \frac{x^3}{2} \, dx
\]
\[
V = \pi \int_{0}^{\sqrt[3]{16}} x^4 \, dx
\]

### Step 4: Solve the integral

Now, integrate:

\[
V = \pi \left[ \frac{x^5}{5} \right]_{0}^{\sqrt[3]{16}}
\]

Substitute the limits of integration:

\[
V = \pi \left( \frac{(\sqrt[3]{16})^5}{5} - \frac{0^5}{5} \right)
\]
\[
V = \pi \cdot \frac{16^{5/3}}{5}
\]

Thus, the volume is:

\[
V = \frac{\pi \cdot 16^{5/3}}{5}
\]

This expression gives the exact volume of the solid of revolution generated by rotating the curve \( y = \frac{x^3}{2} \) around the x-axis.

---

Would you like further details or clarification on this solution?

### 5 Related Questions:
1. What is the physical significance of the cylindrical shells method in finding volume?
2. How would the volume change if the curve was rotated about a different axis?
3. Can we use the disk/washer method for this problem? How would the setup differ?
4. What if the upper limit for \( y \) was a different value, such as \( y = 10 \)?
5. How would the integral change if we rotated the region around the y-axis instead of the x-axis?

### Tip:
Always ensure you correctly identify the radius and height of the cylindrical shell, as these correspond to different components depending on the axis of rotation.

y = x^3/2, y = 8, x = 0 about the x-axis by cylindrical shells

Learn how to calculate the volume of a solid generated by rotating y = x^3/2 around the x-axis using the cylindrical shells method. This advanced calculus problem includes step-by-step integration, finding limits, and applying the cylindrical shells formula.

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