We are tasked with finding the volume of a solid obtained by rotating a region about the $y$-axis. The region is bounded by the curves:

$y = 4\sqrt{x} - 2, \quad y = 0, \quad y = 5$

Part (a) - Definite Integral Setup

Since the rotation is about the $y$-axis, we should express $x$ as a function of $y$. From the equation $y = 4\sqrt{x} - 2$, solve for $x$:

$y + 2 = 4\sqrt{x}$ $\sqrt{x} = \frac{y + 2}{4}$ $x = \left( \frac{y + 2}{4} \right)^2 = \frac{(y + 2)^2}{16}$

The formula for the volume of a solid of revolution about the $y$-axis is given by the method of cylindrical shells:

$V = 2\pi \int_{y_1}^{y_2} x \, dy$

In this case, $x = \frac{(y + 2)^2}{16}$, and the limits of integration for $y$ are from $0$ to $5$. Therefore, the definite integral to find the volume is:

$V = 2\pi \int_0^5 \frac{(y + 2)^2}{16} \, dy$

Part (b) - Compute the Volume

Now, let's compute the integral:

$V = 2\pi \cdot \frac{1}{16} \int_0^5 (y + 2)^2 \, dy$

First, expand $(y + 2)^2$:

$(y + 2)^2 = y^2 + 4y + 4$

Now the integral becomes:

$V = \frac{\pi}{8} \int_0^5 (y^2 + 4y + 4) \, dy$

Now we can compute this integral step by step:

$\int_0^5 y^2 \, dy = \left[ \frac{y^3}{3} \right]_0^5 = \frac{5^3}{3} = \frac{125}{3}$ $\int_0^5 4y \, dy = \left[ 2y^2 \right]_0^5 = 2 \times 25 = 50$ $\int_0^5 4 \, dy = \left[ 4y \right]_0^5 = 4 \times 5 = 20$

Adding all these up:

$V = \frac{\pi}{8} \left( \frac{125}{3} + 50 + 20 \right)$ $V = \frac{\pi}{8} \times \left( \frac{125 + 150 + 60}{3} \right) = \frac{\pi}{8} \times \frac{335}{3}$ $V = \frac{335\pi}{24}$

Now, I'll compute the numerical value of this expression.The volume of the solid obtained by rotating the given region about the $y$-axis is approximately $43.851397$ cubic units, rounded to six decimal places.

Do you have any questions or would you like further details on any part of the solution?

Here are five related questions for further exploration:

1. How would the integral change if the region was rotated around the x-axis instead of the y-axis?
2. What happens if the upper limit of $y$ is changed from 5 to another value?
3. How does the method of cylindrical shells differ from the method of disks or washers?
4. Can you generalize this approach for a function $y = a\sqrt{x} - b$?
5. What if the curve $y = 4\sqrt{x} - 2$ was bounded by different lines?

Tip: When solving problems involving volumes of revolution, always ensure the function is correctly expressed in terms of the variable corresponding to the axis of rotation.