find the volume of the region under the curve f(x)=3x^3 and rotate on the y axis over interval [0,1]

To find the volume of the solid generated by rotating the curve \( f(x) = 3x^3 \) around the **y-axis** over the interval \( [0,1] \), we can use the **method of cylindrical shells**.

The formula for the volume using cylindrical shells is:

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

Where:
- \( f(x) = 3x^3 \)
- \( a = 0 \), \( b = 1 \) (the limits of integration)

### Step 1: Set up the integral
Substitute \( f(x) = 3x^3 \) into the formula:

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

Simplify the expression inside the integral:

\[
V = 2\pi \int_{0}^{1} 3x^4 \, dx
\]

### Step 2: Solve the integral
Now, solve the integral:

\[
V = 2\pi \left[ \frac{3x^5}{5} \right]_{0}^{1}
\]

Evaluate this at the bounds \( x = 1 \) and \( x = 0 \):

\[
V = 2\pi \left( \frac{3(1)^5}{5} - \frac{3(0)^5}{5} \right)
\]

Simplify:

\[
V = 2\pi \times \frac{3}{5}
\]

\[
V = \frac{6\pi}{5}
\]

### Final Answer:
The volume of the solid generated by rotating the curve \( f(x) = 3x^3 \) around the y-axis over the interval \( [0, 1] \) is:

\[
V = \frac{6\pi}{5}
\]

Would you like more details on any of the steps, or have any questions? Here are some related questions:

1. How does the method of cylindrical shells work?
2. Can you explain the difference between cylindrical shells and the disk/washer method?
3. What if we rotated the region around the x-axis instead of the y-axis?
4. What if the interval was different, say [1, 2] instead of [0, 1]?
5. How would we find the surface area of the solid of revolution?

**Tip:** When rotating around different axes, make sure to choose the appropriate method (shell vs. disk/washer) based on the problem's symmetry.

Learn how to calculate the volume of the solid generated by rotating the curve f(x)=3x^3 around the y-axis over the interval [0,1] using the method of cylindrical shells. This calculus problem is suitable for Grades 11-12 and involves integrating the function 3x^3 using the formula for cylindrical shells.

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