To solve the problem, we need to calculate the volume of the solid of revolution using the shell method (metode kulit tabung). Here's how we analyze the situation:

Problem Breakdown:

1. The region is bounded by:

   • $x = y^2$: a parabola opening to the right.
   • $y = 2$: a horizontal line.
   • The solid is rotated around the line $y = 0$.

2. Shell Method Formula: $V = \int_{a}^{b} 2\pi (\text{radius})(\text{height}) \, dy$

   • Radius: The distance from the axis of rotation $y = 0$ is simply $y$.
   • Height: The height of the shell is the horizontal length of the region, which is: $\text{Height} = 4 - y^2$ (from the right boundary $x = 4$ minus $x = y^2$).

3. Limits of Integration: The region extends vertically from $y = 0$ to $y = 2$.

Substituting Values:

$V = \int_{0}^{2} 2\pi (y)(4 - y^2) \, dy$

Simplification:

Expanding the integral: $V = \int_{0}^{2} 2\pi \big( 4y - y^3 \big) \, dy$ This matches Option (e): $V = \int_{0}^{2} 2\pi y (4 - y^2) \, dy$

Correct Answer:

The correct answer is Option (e).

Let me know if you'd like further explanation or detailed steps for integration! Here are five related questions for practice:

1. What is the volume of a solid of revolution if the same region is rotated about $x = 0$?
2. How does the shell method compare to the disk method for this type of problem?
3. What changes in the formula if the axis of rotation is $y = 3$ instead of $y = 0$?
4. Derive the formula for the shell method in general terms.
5. How can you visualize the region being rotated in 3D to better understand the setup?

Tip: Always sketch the region and understand the boundaries before applying integral formulas!