Find the volume *V* of the solid obtained by rotating the region bounded by the given curves about the specified line.

*y* = *x*2,   *y* = 4*x*;

   about the *y*-axis

To find the volume of the solid obtained by rotating the region bounded by the curves \( y = x^2 \) and \( y = 4x \) about the \( y \)-axis, we use the method of cylindrical shells. Here are the steps involved:

1. **Determine the points of intersection:**
   To find the bounds of integration, we first solve for the points where the curves \( y = x^2 \) and \( y = 4x \) intersect.

   Set \( x^2 = 4x \):
   \[
   x^2 - 4x = 0
   \]
   \[
   x(x - 4) = 0
   \]
   So, \( x = 0 \) and \( x = 4 \).

2. **Set up the integral:**
   When using the shell method, the volume \( V \) is given by:
   \[
   V = \int_{a}^{b} 2\pi x (f(x) - g(x)) \, dx
   \]
   where \( f(x) \) is the outer function and \( g(x) \) is the inner function in terms of \( x \). Here, \( f(x) = 4x \) and \( g(x) = x^2 \).

   The bounds are \( x = 0 \) to \( x = 4 \).

3. **Integrate:**
   \[
   V = \int_{0}^{4} 2\pi x (4x - x^2) \, dx
   \]
   Simplify the integrand:
   \[
   V = \int_{0}^{4} 2\pi x (4x - x^2) \, dx = 2\pi \int_{0}^{4} (4x^2 - x^3) \, dx
   \]

   Now, integrate term-by-term:
   \[
   2\pi \int_{0}^{4} (4x^2 - x^3) \, dx = 2\pi \left[ \int_{0}^{4} 4x^2 \, dx - \int_{0}^{4} x^3 \, dx \right]
   \]
   \[
   2\pi \left[ 4 \cdot \frac{x^3}{3} \bigg|_{0}^{4} - \frac{x^4}{4} \bigg|_{0}^{4} \right]
   \]

   Calculate the integrals:
   \[
   2\pi \left[ 4 \cdot \frac{4^3}{3} - \frac{4^4}{4} \right]
   \]
   \[
   2\pi \left[ 4 \cdot \frac{64}{3} - 64 \right]
   \]
   \[
   2\pi \left[ \frac{256}{3} - 64 \right]
   \]
   \[
   2\pi \left[ \frac{256}{3} - \frac{192}{3} \right]
   \]
   \[
   2\pi \left[ \frac{64}{3} \right]
   \]
   \[
   \frac{128\pi}{3}
   \]

Thus, the volume of the solid obtained by rotating the region bounded by \( y = x^2 \) and \( y = 4x \) about the \( y \)-axis is \(\frac{128\pi}{3}\).

Would you like more details on any step or have any questions?

Here are some related questions for further exploration:

1. How would the volume change if the region were rotated about the x-axis instead?
2. What if the region were rotated about the line \( x = 4 \)?
3. How can you use the disk method for this problem?
4. What is the volume if the region is rotated about the line \( y = 4 \)?
5. Can you find the centroid of the region bounded by the curves?
6. How does the method of washers differ from the method of shells?
7. What if the region were bounded by \( y = x^2 \) and \( y = 2x \)?
8. How can numerical methods approximate the volume of more complex regions?

**Tip:** Always check the points of intersection and the setup of the integral carefully to avoid errors in bounds or integrand simplification.

Learn how to calculate the volume of a solid obtained by rotating the region bounded by y = x^2 and y = 4x about the y-axis using the cylindrical shells method. Detailed step-by-step solution provided.

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