Find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines.
y = 5x2
y = 0
x = 3
(a)
y-axis

To find the volume of the solid generated by revolving the region bounded by the given curves around the y-axis, we'll use the **method of cylindrical shells**.

### Given:
- \( y = 5x^2 \)
- \( y = 0 \)
- \( x = 3 \)

### Step-by-Step Solution:

#### 1. **Set up the integral using cylindrical shells:**

The formula for the volume using cylindrical shells when revolving around the **y-axis** is:

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

Here:
- \( f(x) = 5x^2 \)
- The limits of integration are from \( x = 0 \) to \( x = 3 \).

#### 2. **Substitute \( f(x) \) into the formula:**

\[
V = 2\pi \int_{0}^{3} x \cdot 5x^2 \, dx
\]

Simplify the expression inside the integral:

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

#### 3. **Integrate:**

\[
V = 2\pi \left[ \frac{5x^4}{4} \right]_{0}^{3}
\]

Now, evaluate the integral at the limits of integration:

\[
V = 2\pi \left( \frac{5(3)^4}{4} - \frac{5(0)^4}{4} \right)
\]

\[
V = 2\pi \left( \frac{5(81)}{4} \right)
\]

\[
V = 2\pi \left( \frac{405}{4} \right)
\]

\[
V = \frac{405\pi}{2}
\]

Thus, the volume of the solid is:

\[
V = \frac{405\pi}{2} \, \text{cubic units}
\]

### Final Answer:
\[
V = \frac{405\pi}{2} \, \text{cubic units}
\]

Would you like more details on any part of this process?

Here are five related questions to expand on this:
1. How would the volume change if the region was revolved around the line \(x = 4\)?
2. What is the difference between using the disk method and cylindrical shells for this problem?
3. How would you compute the volume if the function was \(y = 5x^3\) instead of \(y = 5x^2\)?
4. What happens if we revolve the region around the line \(x = 2\)?
5. Can you solve this problem by revolving around the x-axis?

**Tip:** When choosing between cylindrical shells and disk/washer methods, cylindrical shells often simplify the integral when revolving around the y-axis if the function is expressed in terms of \(x\).

Find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines.
y = 5x^2
y = 0
x = 3
(a) y-axis

Learn how to find the volume of a solid generated by revolving the region bounded by y = 5x^2, y = 0, and x = 3 around the y-axis using the cylindrical shell method. This detailed step-by-step solution uses calculus and the formula for cylindrical shells. Suitable for students in undergraduate calculus or AP calculus.

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